Math in Focus

Go through the Math in Focus Grade 8 Workbook Answer Key Chapter 2 Lesson 2.1 Understanding Scientific Notation to finish your assignments.

Math in Focus Grade 8 Course 3 A Chapter 2 Lesson 2.1 Answer Key Understanding Scientific Notation

Math in Focus Grade 8 Chapter 2 Lesson 2.1 Guided Practice Answer Key

Tell whether each number is written correctly in scientific notation. If it is incorrectly written, state the reason.

Question 1.
A Brazilian gold frog is about 9.6 • 10° millimeters long.
Answer:
Given,
A Brazilian gold frog is about 9.6 • 10° millimeters long.
9.6 × 1 = 9.6 millimeters
The given statement is correct because the length of a Brazilian gold frog is about 9.6 mm

Question 2.
The wavelength of green light is about 4.15 • 10^-7 meter.
Answer: The wavelength of green light is measured in a nanometer. Thus the above statement is incorrect.

Question 3.
Mars is approximately 0.2244 • 10⁷ kilometers from the Sun.
Answer: The above statement is correct.

Write each number in scientific notation.

Question 4.
856.2
856.2 = • Move the decimal point places to the left and multiply by .
= • Rewrite as a power of 10.
Answer:
856.2 = 8562 • 10^-1 Move the decimal point 1 place to the left and multiply by 8562 • 10^-1
= 8562 • 10^-1

Question 5.
0.06
Answer:
6  • 10^-2 Move the decimal point 2 places to the left and multiply 6 by 10^-2

Write each number in standard form.

Question 6.
9 • 10⁴
9 • 10⁴ = • Evaluate the power.
= Multiply by .
Answer:
9 • 10⁴
• indicates product of two numbers.
9 × 10⁴
9 × 10000 = 90,000
9 multiply by 10000.

Question 7.
2.5 • 10^-2
2.5 • 10^-2 = • Evaluate the power.
= Divide by .
Answer:
2.5 • 10^-2 = 2.5 • 10^-2 Evaluate the power.
= 2.5 Divide by 1/100

Identify the lesser number in each pair of numbers. Justify your reasoning.

Question 8.
4.2 • 10² and 6.5 • 10¹
< Compare the .
<  .
So, is the lesser number.
Answer:
6.5 • 10¹  < 4.2 • 10² Compare the decimals.
6.5 • 10¹  < 4.2 • 10²
So, 6.5 • 10¹ is the lesser number.

Question 9.
3.6 • 10^-3 and 8.4 • 10^-3
The are the same.
< Compare the .
< .
So, is the lesser number.
Answer:
The exponents are the same.
3.6 • 10^-3 < 8.4 • 10^-3
3.6 < 8.4
So, 3.6 is the lesser number

Complete.

Question 10.
An actor has 75,126 fans on a social network. A musician has 8.58 • 104 fans. Who has more fans on the social network?

Method 1
8.58 • 10⁴ =
>
So, the has more fans on the social network.

Method 2
75,126 = •
Compare and .
The are the same.
> Compare the coefficients.
So, the has more fans on the social network.
Answer:
Given,
An actor has 75,126 fans on a social network. A musician has 8.58 • 10⁴ fans.
We have to find who has more fans on the social network.
When you multiply a decimal by a positive power of 10, the decimal point to the right.
8.58 • 10⁴ = 858 × 10²
= 85800 fans
75,126 is less than 85800
Therefore we can say that the musician has more fans on the social network.

Question 11.
The average diameter of a type of round shaped bacteria is 0.0000037 meter. The spacing between two of these bacteria is 2.1 • 10^-9 meter. Which length is shorter?
Answer:
Given,
The average diameter of a type of round shaped bacteria is 0.0000037 meter.
The spacing between two of these bacteria is 2.1 • 10^-9 meter.
0.0000037 meter can be written as
0.37 × 1/100000
0.37 × 10^-5 meter
3.7 ×10 ^-6 meter
37 × 10^-7 meter
2.1 • 10^-9 meter is shorter than 37 • 10^-7 meter

Question 12.
There are 5.816 • 10⁴ spectators in a stadium watching a football game. In a theater, 1,150 people attend an opera. Which venue has more people?
Answer:
Given,
There are 5.816 • 10⁴ spectators in a stadium watching a football game.
In a theater, 1,150 people attend an opera.
We know that,
When you multiply a decimal by a positive power of 10, the decimal point to the right.
5.816 • 10⁴ = 5.816  • 10000
= 5816 × 10
= 58160
1,150
58160 is greater than 1150
Thus the stadium watching a football game has more people than people attend an opera.

Math in Focus Course 3A Practice 2.1 Answer Key

Tell whether each number is written correctly in scientific notation. If incorrectly written, state the reason.

Question 1
71 • 10²²
Answer:
71 • 10²²
No the above number is not written in the scientific notation

Question 2.
8 • 10^-2
Answer:
8 • 10^-2
= 8 × 1/100
= 0.08
No the above number is not written in the scientific notation

Question 3.
0.99 • 10^-3
Answer:
0.99 • 10^-3
= 9.9 × 10^-4
Yes the above number is written in the scientific notation

Question 4.
1.2 • 10^-4
Answer:
1.2 • 10^-4
Yes the above number is written in the scientific notation

Write each number in scientific notation.

Question 5.
533,000
Answer:
We can write the very large number in the standard form as the product of a number between 1 and 10 inclusive of 1 and an integer power of 10.
For the numbers greater than 1 we can use a positive exponent.
533 • 10³

Question 6.
327.8
Answer:
We can write the very large number in the standard form as the product of a number between 1 and 10 inclusive of 1 and an integer power of 10.
For the numbers less than 1 we use a negative exponent.
3278 • 10^-1

Question 7.
0.0034
Answer:
We can write the very large number in the standard form as the product of a number between 1 and 10 inclusive of 1 and an integer power of 10.
For the numbers less than 1 we use a negative exponent.
34 • 10^-4

Question 8.
0.00000728
Answer:
We can write the very large number in the standard form as the product of a number between 1 and 10 inclusive of 1 and an integer power of 10.
For the numbers less than 1 we use a negative exponent.
728 • 10^-8

Write each number in standard form.

Question 9.
7.36 • 10³
Answer: 7360

Question 10.
2.431 • 10⁴
Answer: 24310

Question 11.
5.27 • 10^-2
Answer: 0.0527

Question 12.
4.04 • 10^-4
Answer: 0.000404

Identify the lesser number in each pair of numbers. Justify your reasoning.

Question 13.
8.7 • 10⁶ and 5.9 • 10³
Answer: 5.9 • 10³ < 8.7 • 10⁶

Question 14.
4.8 • 10³ and 9.6 • 10⁷
Answer: 4.8 • 10³ < 9.6 • 10⁷

Question 15.
3.1 • 10^-5 and 7.5 • 10^-5
Answer: 3.1 • 10^-5 < 7.5 • 10^-5

Question 16.
6.9 • 10^-3 and 4.3 • 10^-3
Answer: 4.3 • 10^-3 < 6.9 • 10^-3

Solve. Show your work.

Question 17.
The table shows population data for some countries. Write each population ¡n scientific notation.

Answer:
From the above table, we can write the population in scientific notation.
Brazil – 190,000,000
19 • 10⁷
Singapore – 5,100,000
51 • 10⁵
Monaco – 35,000
35 • 10³
Fiji – 861,000
861 • 10³

Question 18.
Human blood contains red blood cells, white blood cells, and platelets. The table shows the approximate diameters of each of these cells in fractions of a meter. Write each diameter in scientific notation.

Answer:
Given,
Human blood contains red blood cells, white blood cells, and platelets.
The table shows the approximate diameters of each of these cells in fractions of a meter.
Red blood cell – 0.000007
= 7 • 10^-6
White blood cell – 0.00000233
= 233 • 10^-8
Platelet – 0.0000025
= 25 • 10^-7

Question 19.
A praying mantis is approximately 15 centimeters long. A caterpillar is about 76 millimeters long.
a) Write both lengths in millimeters in scientific notation.
b) Write both lengths in centimeters in scientific notation.
How does writing the numbers using the same units help you compare them?
Answer:
a) Write both lengths in millimeters in scientific notation.
Given,
A praying mantis is approximately 15 centimeters long.
15 × 10 millimeters
A caterpillar is about 76 millimeters long.
b) Write both lengths in centimeters in scientific notation.
A praying mantis is approximately 15 centimeters long.
76 × 10^-1 centimeter

Question 20.
A technician reads and records the air pressure from several pressure gauges. The table shows each air pressure reading in pascals (Pa). A pascal is a unit used to measure the amount of force applied on a given area by air or other gases.

a) Which pressure gauge has the greatest reading?
Answer:
Pressure Gauge B has the greatest readings is 5.2 × 10⁵ = 520000

b) Which pressure gauge has the lowest reading?
Answer:
Pressure gauge C has the lowest reading is 170,000

c) The atmospheric pressure when these readings were made was 1.1 • 10⁵ pascals. Which gauge(s) showed a reading greater than the atmospheric pressure?
Answer:
All the gauges A, B, C are showed a greater reading than the atmospheric pressure.
210000 = 2.1 × 10⁵
520000 = 5.2 × 10⁵
170000 = 1.7 × 10⁵

Question 21.
Math Journal The table shows some numbers written in standard form and in the equivalent scientific notation. Describe the relationship between the pair of variables described in a) and b).

a) The value of the positive number in standard form and the sign of the exponent when expressed in scientific notation.
Answer:
When a number is written in scientific notation, the exponent tells you if the term is a large or a small number. A positive exponent indicates a large number and a negative exponent indicates a small number that is between 0 and 1.

b) The sign of the exponent when expressed in scientific notation and the direction the decimal point moves to express the number in standard form.
Answer:
In order to go between scientific notation and decimals, the decimal point is moved to the number of spaces indicated by the exponent. A negative exponent tells you to move the decimal point to the right, while a positive exponent tells you to move it to the left.

Question 22.
Math Journal When visible light passes through a prism the light waves refract, or bend and the colors that make up the light can be seen. Each color has a different wavelength, as shown in the diagram, which is refracted to a different degree.

a) Shorter wavelengths refract more than longer wavelengths. Which color of light wave shows the most refraction? Which color of light wave shows the least refraction?
Answer:
Here the red light shows the longer wavelength is 6.5 × 10^-7, so the red color of the lightwave shows the most reflection. Blue color light shows the least reflection because the wavelength is

b) The frequency of a light wave is the number of waves that travel a given distance in a given amount of time. The shorter the wavelength, the greater the frequency. Order the wavelengths, in order of their frequencies, from least to greatest.
Answer:
Blue: 4.75 • 10^-7
Green: 5.1  • 10^-7
Orange: 5.9  • 10^-7
Red: 6.5 × 10^-7

Go through the Math in Focus Grade 8 Workbook Answer Key Chapter 2 Scientific Notation to finish your assignments.

Math in Focus Grade 8 Course 3 A Chapter 2 Answer Key Scientific Notation

Math in Focus Grade 8 Chapter 2 Quick Check Answer Key

Evaluate.

Question 1.
1.8 • 100
Answer: 180

Explanation:
Here the • indicates the product.
Here the decimal point moves to the right because we are multiplying the decimal number with the whole number 100.
when 1.8 is multiplied by 100 then we get 180.

Question 2.
-0.28 • 10³
Answer: 280

Explanation:
Here the • indicates the product.
Here the decimal point moves to the right because we are multiplying the decimal number with 10³.
-0.28 is multiplied by 10³ gives 280.

Question 3.
1.3 • 10⁴
Answer: 13000

Explanation:
Here the • indicates the product.
Here the decimal point moves to the right because we are multiplying the decimal number with 10⁴.
1.3 is multiplied by 10⁴ gives the result 13000.

Question 4.
74.5 ÷ 1,000
Answer: 0.0745

Explanation:
Here the ÷ indicates the division.
Here the decimal point moves to the left because we are dividing the decimal number by 1000.
74.5 is divided by 1000 gives the result 0.0745.

Question 5.
-3.8 ÷ 10
Answer: 0.38

Explanation:
Here the ÷ indicates the division.
Here the decimal point moves to the left because we are dividing the decimal number by 10.
-3.8 is divided by 10 gives the result -0.38.

Question 6.
2.81 ÷ 10²
Answer: 0.0281

Explanation:
Here the ÷ indicates the division.
Here the decimal point moves to the left because we are dividing the decimal number by 10².
2.81 is divided by 10² gives the result 0.0281.

Go through the Math in Focus Grade 8 Workbook Answer Key Chapter 1 Lesson 1.2 The Product and the Quotient of Powers to finish your assignments.

Math in Focus Grade 7 Course 3 A Chapter 1 Lesson 1.2 Answer Key Exponential Notation

Math in Focus Grade 8 Chapter 1 Lesson 1.2 Guided Practice Answer Key

Simplify each expression. Write your answer in exponential notation.

Question 1.
6⁴ • 6³ =
6⁴ • 6³ = Use the of powers property.
= Simplify.
Answer:
6⁷,

Explanation:
Given 6⁴ • 6³ using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base, So  6⁴ • 6³ = 6^{4 + 3} = 6⁷.

Question 2.
(-5) • (-5)⁵
(-5) • (-5)⁵ = Use the of powers property.
= Simplify the exponent.
= Simplify.
Answer:
(-5)⁶,

Explanation:
Given (-5) • (-5)⁵ using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base, So  (-5) • (-5)⁵ =(-5) ^{1 + 5} = (-5)⁶.

Question 3.

Answer:

Explanation:
Given \(\left(\frac{1}{5}\right)^{3}\) • \(\left(\frac{1}{5}\right)^{4}\)
using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base \(\left(\frac{1}{5}\right)^{3}\) • \(\left(\frac{1}{5}\right)^{4}\) =
\(\left(\frac{1}{5}\right)^{3 +4}\) = \(\left(\frac{1}{5}\right)^{7}\).

Simplify each expression. Write your answer in exponential notation.

Question 4.
p³ • p⁶
p³ • p⁶ = Use the of powers property.
= Simplify.
Answer:
p³ • p⁶ = p^{3 + 6} = p⁹,

Explanation:
Given \({p}^{3}\) • \({p}^{6}\)
using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base \({p}^{3}\) • \({p}^{6}\) =
\({p}^{3 + 6}\) = \({p}^{9}\).

Question 5.
(-c)⁴ • (-c)² 
Answer:
(-c)⁴ • (-c)² = (-c)⁴⁺² = (-c)⁶,

Explanation:
Given \({-c}^{4}\) • \({-c}^{2}\)
using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base \({-c}^{4}\) • \({-c}^{2}\) =
\({-c}^{4 + 2}\) = \({-c}^{6}\).

Question 6.
(3s)⁵ • (3s)
Answer:
(3s)⁵ • (3s) = (3s)⁵⁺¹ = (3s)⁶,

Explanation:
Given \({3s}^{5}\) • \({3s}^{1}\)
using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base \({3s}^{5}\) • \({3s}\) =
\({3s}^{5 + 1}\) = \({3s}^{6}\).

Simplify each expression. Write your answer in exponential notation.

Question 7.
pq³ • p⁵q²

Answer:

Explanation:
Given \({p}{q}^{1}{3}\) • \({p}{q}^{5}{2}\)
using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base \({p}^{1 + 5}\) • \({q}^{3 + 2}\) =
\({p}^{6}\) • \({q}^{5}\).

Question 8.
4s⁴t³ • 5s⁴t⁶

Answer:

Explanation:
Given 4. \({s}{t}^{4}{3}\) • 5 . \({s}{t}^{4}{5}\)
using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base 20 . \({s}^{4+ 4}\) •\({t}^{3 + 6}\) =
20 . \({s}^{8}\) • \({t}^{9}\).

Simplify each expression. Write your answer in exponential notation.

Question 9.
10⁸ ÷ 10⁵
10⁸ ÷ 10⁵ = Use the of powers property.
= Simplify.
Answer:
10⁸ ÷ 10⁵ = 10³,

Explanation:
Given to find 10⁸ ÷ 10⁵  dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take 10⁸ ÷ 10⁵ = 10^8-5= 10³.

Question 10.
2.7⁹ ÷ 2.7⁶ =
= simplify
Answer:
2.7⁹ ÷ 2.7⁶ = 2.7^9-6 = 2.7³,

Explanation:
Given to find 2.7⁹ ÷ 2.7⁶ dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent), means we take 2.7⁹ ÷ 2.7⁶ = 2.7^9-6= 2.7³.

Question 11.
\(\left(\frac{5}{8}\right)^{6}\) ÷ \(\left(\frac{5}{8}\right)\)

= Use the of powers property.
=
Simplify.
Answer:
\(\left(\frac{5}{8}\right)^{5}\),

Explanation:
Given \(\left(\frac{5}{8}\right)^{6}\) ÷ \(\left(\frac{5}{8}\right)\)
dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent) means we take as
\(\left(\frac{5}{8}\right)^{6-1}\) = \(\left(\frac{5}{8}\right)^{5}\).

Simplify each expression. Write your answer in exponential notation.

Question 12.
q⁷ ÷ q²
q⁷ ÷ q² =
= Simplify
Answer:
\({q}^{5}\),

Explanation:
Given to find q⁷ ÷  q²  dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take q⁷ ÷ q² = q^7-2= q⁵.

Question 13.
(-p)⁵ ÷ (-p)³
Answer:
\({-p}^{2}\),

Explanation:
Given to find (-p)⁵ ÷ (-p)²  dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take (-p)⁵ ÷ (-p)³ = (-p)^5-3= (-p)².

Question 14.

Answer:
(r)³ . (s)²,  

Explanation:
Given to find (r)⁸ (s)⁶ ÷ (r)⁵ (s)⁴ dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take
(r)⁸ (s)⁶ ÷ (r)⁵ (s)⁴ = (r)^8-5 . (s)^6-4 = (r)³ . (s)².  

Question 15.

Answer:

Explanation:
Given to find 63 (x)⁹ (y)⁷ ÷ 9.(x)³ (y)⁴ dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take
63 (x)⁹ (y)⁷ ÷ 9 (x)³ (y)⁴ = 7 . (x)^9-3 . (y)^7-4 = 7. (x)⁶ (y)³ .  

Simplify each expression. Write your answer in exponential notation.

Question 16.

Answer:
\({6}^{2}\),

Explanation:
Given to find (6)⁷ (6)³ (6)² ÷ (6)¹ (6)⁴ (6)⁵ dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take
(6)⁷ (6)³ (6)² ÷ (6)¹ (6)⁴ (6)⁵ = (6)⁷⁺³⁺²÷ (6)¹⁺⁴⁺⁵ =
(6)¹² ÷ (6)¹⁰=(6)^12-10 = (6)².

Question 17.

Answer:
\({7.5}^{2}\),

Explanation:
Given to find (7.5)⁵ (7.5)³ (7.5)¹ ÷ (7.5)² (7.5)¹ (7.5)⁴ dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take
(7.5)⁵ (7.5)³ (7.5)¹ ÷ (7.5)² (7.5)¹ (7.5)⁴ = (7.5)⁵⁺³⁺¹÷ (7.5)²⁺¹⁺⁴ =
(7.5)⁹ ÷ (7.5)⁷=(7.5)^9-7 = (7.5)².

Question 18.

Answer:

Explanation:
Given to find (b)⁵ (4a)⁴ (9a)³ ÷ (2a)² (b)² (6a)² dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take
(9).(4).(a)⁴⁺³.(b)⁵ ÷ (2).(6).(a)²⁺² (b)³ = ((36).(a)⁷ .(b)⁵ )÷ (12). (a)⁴.(b)² =
(3)(a)^7-4 (b)^5-2=(3)(a)³ (b)³.

Solve. Show your work.

Question 19.
Jupiter is approximately 108 kilometers from the Sun.
The dwarf planet Eris is about 101° kilometers from the Sun.
How many times as far as Jupiter is Eris from the Sun?

Answer:
Eris is 100 times as far as Jupiter from the Sun,

Explanation:

Math in Focus Course 3A Practice 1.2 Answer Key

Question 1.
(-2)⁶ • (-2)²
Answer:
(-2)⁸ ,

Explanation:
Given (-2)⁶ • (-2)² using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base, So  (-2)⁶ • (-2)² =(-2) ^{6 + 2} = (-2)⁸.

Question 2.
7.²3 • 7.2⁴
Answer:
7.2³ . 7.2⁴ = 7.2³⁺⁴ = 7.2⁷,

Explanation:
Given to find 7.2³ . 7.2⁴ using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base, So 7.2³ . 7.2⁴ = 7.2³⁺⁴= 7.2⁷.

Question 3.
10⁵ • 10⁴
Answer:
10⁹

Explanation:
Given 10⁵ • 10⁴ using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base, So 10⁵ • 10⁴ = 10⁵⁺⁴ = 10⁹.

Question 4.
\(\left(\frac{2}{3}\right) \cdot\left(\frac{2}{3}\right)^{5}\)
Answer:
\(\left(\frac{2}{3}\right)^{6}\),

Explanation:
Given \(\left(\frac{2}{3}\right)^{1}\) • \(\left(\frac{2}{3}\right)^{5}\)
using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base \(\left(\frac{2}{3}\right)^{1}\) • \(\left(\frac{2}{3}\right)^{5}\) =
\(\left(\frac{2}{3}\right)^{1 +5}\) = \(\left(\frac{2}{3}\right)^{6}\).

Question 5.
p • p⁸
Answer:
p¹ • p⁸ = p^{1 + 8} = p⁹,

Explanation:
Given \({p}^{1}\) • \({p}^{8}\)
using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base \({p}^{1}\) • \({p}^{8}\) =
\({p}^{1 + 8}\) = \({p}^{9}\).

Question 6.
q⁸ ÷ q
Answer:
\({q}^{7}\),

Explanation:
Given to find q⁸ ÷  q¹  dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take q⁸ ÷ q¹ = q^8-1= q⁷.

Question 7.
xy² • x⁴y³
Answer:
\({x}^{5}\) • \({y}^{5}\),

Explanation:
Given xy² • x⁴y³
using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base \({x}^{1 + 4}\) • \({y}^{2 + 3}\) =
\({x}^{5}\) • \({y}^{5}\).

Question 8.
2x²y⁴ • 5x⁵y
Answer:
10 X \({x}^{7}\) • \({y}^{5}\),

Explanation:
Given 2x²y⁴ • 5x⁵y using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base (2 X 5) X \({x}^{2 + 5}\) • \({y}^{4 + 1}\) =
10 X \({x}^{7}\) • \({y}^{5}\).

Question 9.
2.5x³y⁶ • 3x²y⁴
Answer:
7.5 X \({x}^{5}\) • \({y}^{10}\),

Explanation:
Given 2.5 X x³y⁶ • 5x²y⁴ using the Product of Powers Property states that when
multiplying two exponents with the same base, we add the exponents and
keep the base (2.5 X 3) X \({x}^{3 + 2}\) • \({y}^{6 + 4}\) =
7.5 X \({x}^{5}\) • \({y}^{10}\).

Question 10.
(-3)⁴ ÷ (-3)²
Answer:
(-3)² ,

Explanation:
Given to find (-3)⁴ ÷  (-3)²  dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take (-3)⁴ ÷ (-3)² = (-3)^4-2= (-3)².

Question 11.
2¹⁰ ÷ 2⁵
Answer:
2⁵,

Explanation:
Given to find 2¹⁰ ÷  2⁵  dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take 2¹⁰ ÷ 2⁵ = 2^10-5= 2⁵.

Question 12.
\(\left(-\frac{1}{6}\right)^{5} \div\left(-\frac{1}{6}\right)^{2}\)
Answer:
\(\left(-\frac{1}{6}\right)^{3}\),

Explanation:
Given \(\left(-\frac{1}{6}\right)^{5}\) ÷ \(\left(-\frac{1}{6}\right)^{2}\)
dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent) means we take as
\(\left(-\frac{1}{6}\right)^{5-2}\) = \(\left(-\frac{1}{6}\right)^{3}\).

Question 13.
63y³z⁵ ÷ 9
Answer:
7y³z⁵,

Explanation:
Given to find 63 (y)³ (z)⁵ ÷ 9 dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take
63 (y)³ (z)⁵ ÷ 9 = 7. (y)³ (z)⁵ .  

Question 14.
h²k⁵ ÷ hk⁴
Answer:
hk,

Explanation:
Given to find  (h)² (k)⁵ ÷ h.(k)⁴ dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take
(h)² (k)⁵ ÷  h. (k)⁴ = (h)^2-1 . (k)^5-4 = hk.  

Question 15.
64a⁸b⁵ ÷ 4a³b²
Answer:
16a⁵b³,

Explanation:
Given to find 64 (a)⁸ (b)⁵ ÷ 4.(a)³ (b)² dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take
64 (a)⁵ ( b)³ ÷ 4 (a)³ (b)² = 16 . (a)^8-3 . (b)^5-2 = 16. a⁵b³ . 

Question 16.
\(\frac{5^{9} \cdot 5^{7} \cdot 5^{8}}{5^{3} \cdot 5^{2} \cdot 5}\)
Answer:
5¹⁸,

Explanation:
Given to find (5)⁹ (5)⁷ (5)⁸ ÷ (5)³ (5)² (5)¹ dividing them with the exponent value in
(5)⁹ (5)⁷ (5)⁸ ÷ (5)³ (5)² (5)¹ the numerator (the top exponent) and
subtract the exponent value of the denominator (the bottom exponent).
Here that means we take (5)⁹⁺⁷⁺⁸ ÷ (5)³⁺²⁺¹ = (5)²⁴ ÷ (5)⁶=(5)^24-6 = (5)¹⁸.

Question 17.

Answer:
\(\left(\frac{4}{9}\right)^{5}\),

Explanation:
Given to find \(\left(\frac{4}{9}\right)^{6}\). \(\left(\frac{4}{9}\right)^{5}\).
\(\left(\frac{4}{9}\right)^{4}\)  ÷ \(\left(\frac{4}{9}\right)^{3}\). \(\left(\frac{4}{9}\right)^{3}\). \(\left(\frac{4}{9}\right)^{4}\) dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take
\(\left(\frac{4}{9}\right)^{6+5+4}\) ÷ \(\left(\frac{4}{9}\right)^{3 +3+4}\),
\(\left(\frac{4}{9}\right)^{15}\) ÷ \(\left(\frac{4}{9}\right)^{10}\),
\(\left(\frac{4}{9}\right)^{15-10}\) = \(\left(\frac{4}{9}\right)^{5}\).

Question 18.
\(\frac{a^{9} \cdot a^{2} \cdot a^{3}}{a^{6} \cdot a^{3} \cdot a^{4}}\)
Answer:
a,

Explanation:
Given \(\frac{a^{9} \cdot a^{2} \cdot a^{3}}{a^{6} \cdot a^{3} \cdot a^{4}}\)
to find dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take
\(\frac{a^{9+2+3} \cdot}{a^{6+3+4} \cdot}\) =
\(\frac{a^{14} \cdot}{a^{13} \cdot}\) = (a)^14-13 = = a.

Question 19.
\(\frac{b^{4} \cdot b^{6} \cdot b}{b^{3} \cdot b^{3} \cdot b^{3}}\)
Answer:
(b)²,

Explanation:
Given \(\frac{b^{4} \cdot b^{6} \cdot b}{b^{3} \cdot b^{3} \cdot b^{3}}\)
to find dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take
\(\frac{b^{4+6+1} \cdot}{a^{3+3+3} \cdot}\) =
\(\frac{b^{11} \cdot}{b^{9} \cdot}\) = (b)^11-9 = (b)².

Question 20.
\(\frac{3 x^{3} \cdot z^{4} \cdot 4 x^{3}}{2 x \cdot x \cdot 3 z}\)
Answer:
2x⁴z³,

Explanation:
Given to find 3.(x)³ (z)⁴ . 4(x)³ ÷ 2.(x)(x) (z) dividing them with the exponent value in
the numerator (the top exponent) and subtract the exponent value of the
denominator (the bottom exponent). Here that means we take
12 (x)⁶ (z)⁴ ÷ 6 (x)² (z) = 2 . (x)^6-2 . (z)^4-1 = 2. (x)⁴ (z)³ .  

Question 21.
\(\frac{4 c^{6} \cdot 3 b^{4} \cdot 9 c^{5}}{b^{3} \cdot 6 c^{3} \cdot 2 c^{3}}\)
Answer:
9bc⁵,

Explanation:
Given to find \(\frac{4 c^{6} \cdot 3 b^{4} \cdot 9 c^{5}}{b^{3} \cdot 6 c^{3} \cdot 2 c^{3}}\)
dividing them with the exponent value in the numerator (the top exponent) and
subtract the exponent value of the denominator (the bottom exponent).
Here that means we take 4(c)⁶. 3(b)⁴ . 9(c)⁵  ÷ (b)³ . 6 (c)³ . 2(c)³  =
9 . (b)^4-3 . (c)^11-6 = 9.(b).(c)⁵ .  

Solve. Show your work.

Question 22.
Pluto has a diameter of about 103 kilometers. The diameter of Saturn is
approximately 105 kilometers.
How many times as great as Pluto’s diameter is Saturn’s diameter?

Answer:
100 times as great as Pluto’s diameter is Saturn’s diameter,

Explanation:
Given Pluto has a diameter of about (10)³ kilometers.
The diameter of Saturn is approximately (10)⁵ kilometers.
So number of times as great as Pluto’s diameter is Saturn’s diameter is
(10)^5-3 = (10)²,
therefore 100 times as great as Pluto’s diameter is Saturn’s diameter.

Question 23.
Use the rectangular prism shown.
a) Express the volume of the rectangular prism using exponential notation.
Answer:
6(x)³,

Explanation:
Given width of the rectangular prism as 2x units,
height as x units and length as 3x units so the
volume of the rectangular prism using exponential notation is
w X h X l = 2x X x X 3x =6(x)¹⁺¹⁺¹ = 6(x)³.

b) Another prism has dimensions that are twice the dimensions of the prism shown.
Express the volume of that prism using exponential notation.
Answer:

Explanation:
Given another prism has dimensions that are twice the dimensions of the prism shown.
Expressing the volume of that prism using exponential notation is
width of the rectangular prism as 2x X 2x = 4x¹⁺¹  = 4x² units,
height as 2 X x = 2x units and length as 3x X 3 x = 9x¹⁺¹ = 9x² units,
So the volume of the rectangular prism using exponential notation is
w X h X l = 4x² X 2x X 9x² = 72 (x)²⁺¹⁺² = 72(x)⁵.

c) How many times greater is the volume of the larger prism than the volume of the smaller prism?
Answer:
12x² times greater is the volume of the larger prism than the volume of the smaller prism,

Explanation:
To know number of times greater is the volume of the larger prism than the
volume of the smaller prism, we divide 72(x)⁵ ÷ 6(x)³ = 12(x)^5-3 = 12(x)².

Go through the Math in Focus Grade 8 Workbook Answer Key Chapter 1 Lesson 1.3 The Power of a Power to finish your assignments.

Math in Focus Grade 7 Course 3 A Chapter 1 Lesson 1.3 Answer Key The Power of a Power

Math in Focus Grade 8 Chapter 1 Lesson 1.3 Guided Practice Answer Key

Hands-On Activity

Materials:

• three stacks of five cards, each labeled with a number from

Explore The Power

In this activity, you and your partner will play a game in which you write and
evaluate expressions in the form (a^m)ⁿ.
You will get a point for each expression you write, and the person with the greater score wins.

Step 1.
Shuffle each stack of number cards and place them in a pile. Each player randomly draws three cards, one from each pile.

Step 2
Use your three cards to write an expression in the form (a^m)ⁿ.
For instance, if you draw 2, 4, and 5, you could write (2⁴)⁵,(4²)⁵, or another expression.
Write as many expressions as you can. You may want to use a calculator to evaluate your expression.
For instance, to evaluate (2⁴)⁵, you can use these keystrokes:

Step 3.
Record your expressions and their values.
Your partner should also record his or her expressions and their values.
Check your partner’s work.

Step 4.
Continue the game by replacing the cards you used and shuffling the piles.
Repeat Step 1 to Step 2 several more times. When you are finished,
find each player’s score by counting the number of correct expressions that
each player has written. The player with the greater score wins.

Math Journal Is it correct to assume that using the greatest number drawn as
the base will give the expression with the greatest possible value?
Explain or give an example.
Answer:
Yes,
Example: Drawing numbers 7,3,8,

Explanation:
Yes it is correct to assume that using the greatest number drawn as
the base will give the expression with the greatest possible value example
if we draw 7,3,8 we could write (8⁷)³,(7⁸)³.

Simplify each expression. Write your answer in exponential notation.

Question 1.
(5³)⁴
(5³)⁴ = Use the of a power property.
= Simplify.
Answer:
(5)¹²,

Explanation:
Given (5³)⁴ used the power of a power property and simplified as
(5)^3X4 = (5)¹².

Question 2.
(2.3⁴)²
(2.3⁴)² = Use the of a power property.
= Simplify.
Answer:
(2.3)⁸,

Explanation:
Given (2.3⁴)² used the power of a power property and simplified as
(2.3)^{4 X 2} = (2.3)⁸.

Question 3.
[(3p)⁵]⁴
Answer:
(3p)²⁰,

Explanation:
Given [(3p)⁵]⁴ used the power of a power property and simplified as
(3p)^{5 X 4} = (3p)²⁰.

Question 4.
[(-y)⁴]⁷
Answer:
(-y)²⁸,

Explanation:
Given (-y⁴)⁷ used the power of a power property and simplified as
(-y)^4X7 = (-y)²⁸.

Simplify each expression. Write your answer in exponential notation.

Question 5.
[(-3) • (-3)⁶]²

Answer:
(-3)¹⁴,

Explanation:

Question 6.
[p⁴ • p²]⁵

Answer:
(p)³⁰,

Explanation:

Question 7.
(6³ • 6³]⁷ ÷ 6¹⁰

Answer:
(6)³²,

Explanation:

Question 8.

Answer:
(x)⁶,

Explanation:

Math in Focus Course 3A Practice 1.3 Answer Key

Simplify each expression. Write your answer in exponential notation.

Question 1.
(2⁶)²
Answer:
(2)¹²,

Explanation:
Given (2⁶)² used the power of a power property and simplified as
(2)^{6 X 2} = (2)¹².

Question 2.
(3⁴)³
Answer:
(3)¹²,

Explanation:
Given (3⁴)³ used the power of a power property and simplified as
(3)^{4 X 3} = (3)¹².

Question 3.
(10⁵)⁴
Answer:
(10)²⁰,

Explanation:
Given (10⁵)⁴ used the power of a power property and simplified as
(10)^{5 X 4} = (10)²⁰.

Question 4.
(10⁷)²
Answer:
(10)¹⁴,

Explanation:
Given (10⁷)² used the power of a power property and simplified as
(10)^{7X 2} = (10)¹⁴.

Question 5.
(25³)³
Answer:
(25)⁹,

Explanation:
Given (25³)³ used the power of a power property and simplified as
(25)^{3X 3} = (25)⁹.

Question 6.
(x⁶)³
Answer:
(x)¹⁸,

Explanation:
Given (x⁶)³ used the power of a power property and simplified as
(x)^{6 X 3} = (x)¹⁸.

Question 7.
\(\left[\left(\frac{1}{8}\right)^{3}\right]^{6}\)
Answer:
\([\left(\frac{1}{8}\right)^{18}\),

Explanation:
Given \([\left(\frac{1}{8}\right)^{3}]^{6}\)
used the power of a power property and simplified as
\([\left(\frac{1}{8}\right)^{3 X 6}\) =
\([\left(\frac{1}{8}\right)^{18}\).

Question 8.
\(\left[\left(\frac{4}{5}\right)^{2}\right]^{4}\)
Answer:
\([\left(\frac{4}{5}\right)^{8}\),

Explanation:
Given \(\left[\left(\frac{4}{5}\right)^{2}\right]^{4}\)
used the power of a power property and simplified as
\([\left(\frac{4}{5}\right)^{2X4}\) =
\([\left(\frac{4}{5}\right)^{8}\).

Question 9.
[(2y)³]⁸
Answer:
(2y)²⁴,

Explanation:
Given [(2y)³)]⁸ used the power of a power property and simplified as
(2y)^3X8 = (2y)²⁴.

Question 10.
[(57p)⁴]⁴
Answer:
(57p)¹⁶,

Explanation:
Given [(57p)⁴)⁴ used the power of a power property and simplified as
(57p)^{4 X 4} = (57)¹⁶.

Question 11.
[(-6)⁴]³
Answer:
(-6)¹²,

Explanation:
Given [(-6)⁴)³ used the power of a power property and simplified as
(-6)^4X3 = (-6)¹².

Question 12.
[(-p)²]¹¹
Answer:
(-p)²²,

Explanation:
Given [(-p)²]¹¹ used the power of a power property and simplified as
(-p)^2X11 = (-p)²².

Question 13.
Math Journal Michael thinks that (a³)² = a⁵. Is he correct? Why?

Answer:
No, Michael is not correct as [(a)³]² = (a)⁶≠(a)⁵,

Explanation:
Given (a³)² = a⁵ used the power of a power property and simplified as
(a)^3X2 = (a)⁶≠(a)⁵. Therefore Michael is incorrect.

Simplify each expression. Write your answer in exponential notation.

Question 14.
(5⁵ • 5⁶)²
Answer:
(5)²²,

Explanation:
Given ((5⁵ • 5⁶)² used the product of a power property as
(5⁵⁺⁶)² = (5¹¹)² and using power of power property simplified as
(5)^11X2 = (5)²².

Question 15.
(p⁴ • p²)⁶
Answer:
(p)⁴⁸,

Explanation:
Given ((5⁵ • 5⁶)² used the product of a power property as
(5⁵⁺⁶)² = (5¹¹)² and using power of power property simplified as
(5)^11X2 = (5)²².

Question 16.
\(\left[\left(\frac{1}{2}\right) \cdot\left(\frac{1}{2}\right)^{3}\right]^{5}\)
Answer:
\(\left[(\frac{1}{2}\right)^{20}\),

Explanation:
Given \(\left[\left(\frac{1}{2}\right) \cdot\left(\frac{1}{2}\right)^{3}\right]^{5}\)
Using the product of a power property as \(\left[\left(\frac{1}{2}\right)^{1+
3}\right]^{5}\) = \(\left[\left(\frac{1}{2}\right)^{4}\right]^{5}\) and using power of power property simplified as \(\left(\frac{1}{2}\right)^{4 X 5}\) = \(\left(\frac{1}{2}\right)^{20}\).

Question 17.
\(\left[\left(-\frac{4}{9}\right)^{2} \cdot\left(-\frac{4}{9}\right)^{3}\right]^{2}\)
Answer:
\(\left[(-\frac{4}{9}\right)^{10}\),

Explanation:
Given \(\left[\left(-\frac{4}{9}\right)^{2} \cdot\left(-\frac{4}{9}\right)^{3}\right]^{2}\)
Using the product of a power property as \(\left[\left(-\frac{4}{9}\right)^{2+
3}\right]^{2}\) = \(\left[\left(-\frac{4}{9}\right)^{5}\right]^{2}\) and using power of power property simplified as \(\left(-\frac{4}{9}\right)^{5 X 2}\) = \(\left(-\frac{4}{9}\right)^{10}\).

Question 18.
(2² • 2⁴)³ ÷ 2⁸
Answer:
2¹⁰,

Explanation:
Given (2² • 2⁴)³ ÷ 2⁸ used the product of a power property as
(2²⁺⁴)³  ÷ 2⁸  = (2⁶)³ ÷ 2⁸ and using power of power property simplified as
(2)^6X3 ÷ 2⁸  = ( 2)¹⁸ ÷2⁸ using division of power property ( 2)^18-8 = (2)¹⁰.

Question 19.
(7 • 7²)⁵ ÷ 7³
Answer:
^{( 7)12,}

Explanation:
Given (7 • 7²)⁵ ÷ 7³ used the product of a power property as
(7¹⁺²)⁵  ÷ 7³  = (7³)⁵ ÷ 7³ and using power of power property simplified as
(7)^3X5 ÷ 7³  = ( 7)¹⁵ ÷7³ using division of power property ( 7)^15-3 = ( 7)¹².

Question 20.
(s⁶ • s²) ÷ s⁴
Answer:
(s)⁴,

Explanation:
Given (s⁶ • s²) ÷ s⁴ used the product of a power property as
(s⁶⁺²) ÷ s⁴  = (s⁸) ÷ s⁴  using division of power property ( s)^8-4 = ( s)⁴.

Question 21.
(t⁴ • t⁴) ÷ t⁴
Answer:
(t)⁴,

Explanation:
Given (t⁴ • t⁴) ÷ t⁴ used the product of a power property as
(t⁴⁺⁴) ÷ t⁴  = (t⁸) ÷ t⁴  using division of power property (t)^8-4 = ( t)⁴.

Question 22.
\(\frac{\left(8^{8} \cdot 8^{3}\right)^{2}}{\left(8^{5}\right)^{4}}\)
Answer:
8²,

Explanation:
Given \(\frac{\left(8^{8} \cdot 8^{3}\right)^{2}}{\left(8^{5}\right)^{4}}\)  used the product of a power property as \(\frac{\left(8^{8+3}\right)^{2}}{\left(8^{5}\right)^{4}}\)
and using power of power property simplified as \(\frac{(8)^{11 X 2}}{(8)^{5 X 4}}\)
 using division of power property (8)^22-20 = (8)².

Question 23.
\(\frac{\left(3^{4} \cdot 3^{2}\right)^{4}}{\left(3^{5}\right)^{2}}\)
Answer:
3¹⁴,

Explanation:
Given \(\frac{\left(3^{4} \cdot 3^{2}\right)^{4}}{\left(3^{5}\right)^{2}}\) used the product of a power property as \(\frac{\left(3^{4+2}\right)^{4}}{\left(3^{5}\right)^{2}}\)
and using power of power property simplified as \(\frac{(3)^{6 X 4}}{(3)^{5 X 2}}\)
 using division of power property (3)^24-10 = (3)¹⁴.

Question 24.
\(\frac{\left(b \cdot b^{3}\right)^{5}}{\left(b^{2}\right)^{4}}\)
Answer:
b¹²,

Explanation:
Given \(\frac{\left(b \cdot b^{3}\right)^{5}}{\left(b^{2}\right)^{4}}\) used the product of a power property as \(\frac{\left(b^{1+3}\right)^{5}}{\left(b^{2}\right)^{4}}\)
and using power of power property simplified as \(\frac{(b)^{4 X 5}}{(b)^{2X 4}}\)
 using division of power property (b)^20-8 = (b)¹².

Question 25.
\(\frac{\left(h^{6} \cdot h^{6}\right)^{2}}{\left(h^{3}\right)^{5}}\)
Answer:
h⁹,

Explanation:
Given \(\frac{\left(h^{6} \cdot h^{6}\right)^{2}}{\left(h^{3}\right)^{5}}\) used the product of a power property as \(\frac{\left(h^{6+6}\right)^{2}}{\left(h^{3}\right)^{5}}\)
and using power of power property simplified as \(\frac{(h)^{12 X 2}}{(h)^{3X 5}}\)
 using division of power property (h)^24-15 = (h)⁹.

Question 26.
(q⁵ • q²)³) ÷ 5q⁵
Answer:
(q)¹⁶÷ 5 or 1/5 . q¹⁶,

Explanation:
Given ((q⁵ • q²)³) ÷ 5q⁵  used the product of a power property as
((q⁵⁺²)³) ÷ 5q⁵  and using power of power property simplified as ((q)^{7 X 3} ) ÷ 5q⁵  using
division of power property (q)^21-5 ÷ 5 = (q)¹⁶÷ 5 or 1/5 . q¹⁶.

Question 27.
(c⁷ • q³)⁴ ÷ 6c²
Answer:
(c²⁶• q¹²) ÷ 6 or 1/6 . q¹⁶,

Explanation:
Given ((c⁷ • q³)⁴) ÷ 6c² using power of power property simplified as
((c)^{7 X4} • q^3X4) ÷ 6c²  using division of power property ((c)^28-2 • q¹² ) ÷ 6 =
(c²⁶• q¹²) ÷ 6 or 1/6 . q¹⁶.

Question 28.

Answer:
\(\left(\frac{2}{3}\right)^{2}\),

Explanation:
Given (\(\left[(\frac{2}{3}\right)^{2}\) . \(\left[(\frac{2}{3}\right)^{6}\))÷\(\left[(\frac{2}{3}\right)^{2 X 3}\)] using product of power property simplified as
\(\left(\frac{2}{3}\right)^{2+ 6}\) ÷\(\left(\frac{2}{3}\right)^{6}\) using
division of power property \(\left[(\frac{2}{3}\right)^{8-6}\)] we get
\(\left(\frac{2}{3}\right)^{2}\).

Question 29.

Answer:
\(\left(\frac{x}{2}\right)^{1}\),

Explanation:
Given (\(\left[(\frac{x}{2}\right)^{3}\) . \(\left[(\frac{x}{2}\right)^{4}\))÷\(\left[(\frac{x}{2}\right)^{3 X 2}\)] using product of power property simplified as
\(\left(\frac{x}{2}\right)^{3+ 4}\) ÷\(\left(\frac{x}{2}\right)^{6}\) using
division of power property \(\left[(\frac{x}{2}\right)^{7-6}\)] we get
\(\left(\frac{x}{2}\right)^{1}\).

Go through the Math in Focus Grade 8 Workbook Answer Key Chapter 1 Exponents to finish your assignments.

Math in Focus Grade 8 Course 3 A Chapter 1 Answer Key Exponents

Math in Focus Grade 8 Chapter 1 Quick Check Answer Key

Identify the integers in the list below.

Question 1.
4.31, -6, 8.12, 58, -211,43
Answer:
-211,-6,43 and 58,

Explanation:
Integers are positive,negative and zero but not decimal numbers therefore
from the given numbers 4.31 and 8.12 are not integers
the integers are -211,-6,43 and 58.

Write each rational number as a terminating decimal.

Question 2.
\(\frac{3}{5}\)
Answer:
0.6,

Explanation:

when we divide 3 by 5 we get terminating decimal as 0.6.

Question 3.
4\(\frac{1}{4}\)
Answer:
4.25,

Explanation:
Given 4\(\frac{1}{4}\) we get \(\frac{4 x 4 + 1}{4}\) =
\(\frac{17}{4}\)
the terminating decimal is 4.25.

Write each rational number as a repeating decimal.

Question 4.
\(\frac{6}{11}\)
Answer:
The repeating decimal of 6/11 is 0.5363636 and so on,

Explanation:
Given to find \(\frac{6}{11}\) we get

the repeating decimal of 6/11 is 0.5363636 and so on.

Question 5.
\(\frac{5}{12}\)
Answer:
The repeating decimal of 5/12 is 0.4166 so on,

Explanation:

The repeating decimal of 5/12 is 0.4166 so on.

Locate each irrational number on a number line using rational approximation.

Question 6.

Answer:

Explanation:
Located irrational number cube root of -47 on the
number line – 3.6 using rational approximation.

Question 7.
\(\sqrt{19}\)
Answer:

Explanation:
Located irrational number square root of 19 on the
number line 4.35 using rational approximation.

Evaluate each expression.

Question 8.
-3 + (-4)
Answer:
-7,

Explanation:
Adding  -3 to -4 we get -7.

Question 4.
-4 – (-2)
Answer:
-6,

Explanation:
Adding -4 to -2 we get -6.

Evaluate each expression.

Question 10.
(-7) • (-3)
Answer:
21,

Explanation:
Given -7 multiplied by -3 we get – 7 X -3 = 21.

Question 11.
(-12) ÷ 3
Answer:
-4,

Explanation:
Given to divide (-12) with 3 we get -4.

Go through the Math in Focus Grade 8 Workbook Answer Key Chapter 1 Lesson 1.4 The Power of a Product and the Power of a Quotient to finish your assignments.

Math in Focus Grade 7 Course 3 A Chapter 1 Lesson 1.4 Answer Key The Power of a Product and the Power of a Quotient

Math in Focus Grade 8 Chapter 1 Lesson 1.4 Guided Practice Answer Key

Simplify each expression. Write your answer in exponential notation.

Question 1.
6³ • 7³
6³ • 7³ = Use the power of a property.
= Simplify.
Answer:
(6 X 7)³ = (42)³,

Explanation:
Given 6³ • 7³  when the variable bases are different and the
powers are the same, the bases are multiplied first,
Used the powers of product property,
(6 X 7)³ = (42)³.

Question 2.

Answer:
(5/24)⁴,

Explanation:
Given (-5/6)⁴. (- 1/4)⁴ when the variable bases are different and the
powers are the same, the bases are multiplied first,
Used the powers of product and quotients of a property,
(-5/6 X -1/4)⁴ = (5/24)⁴.

Question 3.
(1.8)² • (0.75)²
(1.8)² • (0.75)² = Use the power of a property.
= Simplify.
Answer:
((1.8) • (0.75))²,

Explanation:
Given (1.8)² • (0.75)² when the variable bases are different and the
powers are the same, the bases are multiplied first,
Used the powers of product of a property,
((1.8) • (0.75))².

Simplify each expression. Write your answer in exponential notation.

Question 4.
p⁶ • q⁶
p⁶ • q⁶ = Use the power of a property.
= Simplify.
Answer:
(p X q)⁶ ,

Explanation:
Given p⁶ • q⁶  when the variable bases are different and the
powers are the same, the bases are multiplied first,
Used the powers of product property,
(p X q)⁶ .

Question 5.
(3a)⁴ • (4b)⁴
(3a)⁴ • (4b)⁴ = Use the power of a property.
= Simplify.
Answer:
(3a X 4b)⁴ ,

Explanation:
Given (3a)⁴ • (4b)⁴  when the variable bases are different and the
powers are the same, the bases are multiplied first,
Used the powers of product property we get (3aX 4b)⁴.

Question 6.
(-3y²)³ • \(\left(\frac{1}{12 y}\right)^{3}\)
Answer:
\(\left(-\frac{y}{4}\right)^{3}\),

Explanation:
Given (-3y²)³ • \(\left(\frac{1}{12 y}\right)^{3}\) on simplification
(-y)^{2 X 3} • \(\left(\frac{1}{4 y}\right)^{3}\) = (-y)^6-3 • \(\left(\frac{1}{4}\right)^{3}\) =
(-y)³ • \(\left(\frac{1}{4}\right)^{3}\) = \(\left(-\frac{y}{4}\right)^{3}\).

Simplify each expression. Write your answer in exponential notation.

Question 7.
2⁵ ÷ 4⁵

Answer:
\(\left(\frac{1}{2}\right)^{5}\),

Explanation:
Given 2⁵ ÷ 4⁵  as 4 can be written as (2 X 2) so 2⁵ ÷ (2 X 2)⁵ upon simplification
we get 1/2⁵ or \(\left(\frac{1}{2}\right)^{5}\).

Question 8.
(-9)³ ÷ (-3)³

Answer:
(3)³,

Explanation:
Given(-9)³ ÷ (-3)³  as (-9) can be written as (-3 X 3) so (-3 X 3)³ ÷ (-3)³
on simplification we get (3)³.

Simplify each expression. Write your answer in exponential notation.

Question 9.
x⁴ ÷ y⁴

Answer:
\(\left(\frac{x}{y}\right)^{4}\),

Explanation:
Given x⁴ ÷ y⁴ when the variable bases are different and the
powers are the same, the bases are simplified first as (x ÷ y)⁴ and solved.

Question 10.
(8p)⁵ ÷ (3q)⁵
Answer:
(8p ÷ 3q)⁵,

Explanation:
Given (8p)⁵ ÷ (3q)⁵ when the variable bases are different and the
powers are the same, the bases are simplified first as (8p ÷ 3q)⁵ and solved.

Simplify each expression. Write your answer in exponential notation.

Question 11.

Answer:
(2)⁷,

Explanation:

Question 12.

Answer:
(4)³ X (3)³,

Explanation:

Question 13.

Answer:
(6)⁸,

Explanation:

Question 14.
\(\frac{2^{5} \cdot 3^{12} \cdot 2^{7}}{6^{7}}\)
Answer:
(6)⁵,

Explanation:
Given \(\frac{2^{5} \cdot 3^{12} \cdot 2^{7}}{6^{7}}\),
6 can be written as 2 X 3 so \(\frac{2^{5} \cdot 3^{12} \cdot 2^{7}}{({2 X 3})^{7}}\),

Question 15.
\(\frac{\left(25^{3}\right)^{2} \cdot 7^{6}}{5^{6}}\)
Answer:
(35)⁶,

Explanation:
Given \(\frac{\left(25^{3}\right)^{2} \cdot 7^{6}}{5^{6}}\)

Math in Focus Course 3A Practice 1.4 Answer Key

Simplify each expression. Write your answer in exponential notation.

Question 1.
5⁴ • 6⁴
Answer:
(5 X 6)⁴ = (30)⁴,

Explanation:
Given 5⁴ • 6⁴  when the variable bases are different and the
powers are the same, the bases are multiplied first,
Used the powers of product property,
(5 X 6)⁴ = (30)⁴.

Question 2.
5.4³ • 4.5³
Answer:
(5.4 X 4.5)³ = (24.3)³,

Explanation:
Given 5.4³ • 4.5³  when the variable bases are different and the
powers are the same, the bases are multiplied first,
Used the powers of product property,
(5.4 X 4.5)³ = (24.3)³.

Question 3.
2⁵ • 10⁵
Answer:
(2 X 10)⁵ = (20)⁵,

Explanation:
Given 2⁵ • 10⁵  when the variable bases are different and the
powers are the same, the bases are multiplied first,
Used the powers of product property,
(2 X 10)⁵ = (20)⁵.

Question 4.
a³ • b³
Answer:
(a X b)³ = (ab)³,

Explanation:
Given a³ • b³  when the variable bases are different and the
powers are the same, the bases are multiplied first,
Used the powers of product property,
(a X b)³ = (ab)³.

Question 5.
(2x)⁵ • (3y)⁵
Answer:
(2x X 3y)⁵ = (6xy)⁵,

Explanation:
Given (2x)⁵ • (3y)⁵  when the variable bases are different and the
powers are the same, the bases are multiplied first,
Used the powers of product property,
(2x X3y)⁵ = (6xy)⁵.

Question 6.
(2.5a)⁶ • (1.6b)⁶
Answer:
(2.5a X 1.6b)⁶ = (4ab)⁶,

Explanation:
Given (2.5a)⁶ • (1.6b)⁶  when the variable bases are different and the
powers are the same, the bases are multiplied first,
Used the powers of product property,
(2.5a X 1.6b)⁶ = (4ab)⁶.

Question 7.
\(\left(-\frac{1}{3}\right)^{4} \cdot\left(-\frac{2}{5}\right)^{4}\)
Answer:
\(\left(\frac{2}{15}\right)^{4}\),

Explanation:
Given \(\left(-\frac{1}{3}\right)^{4} \cdot\left(-\frac{2}{5}\right)^{4}\)
when the variable bases are different and the powers are the same,
the bases are multiplied first,Used the powers of product property,
\(\left(\frac{-1 X -2}{3X 5}\right)^{4}\) = \(\left(\frac{2}{15}\right)^{4}\).

Question 8.
9² ÷ 3²
Answer:
(3)²,

Explanation:
Given (9)² ÷ (3)²  as 9 = 3 X 3 can be written as (3²)² so (3)⁴ ÷ (3)²
on simplification we get (3)^4-2 = (3)².

Question 9.
10⁶ ÷ 5⁶
Answer:
(2)⁶,

Explanation:
Given (10)⁶ ÷ (5)⁶  as 10 can be written as (5 x 2)⁶ so (5)⁶ .(2)⁶ ÷ (5)⁶
on simplification we get (2)⁶.

Question 10.
2.8⁷ ÷ 0.7⁷
Answer:
(4)⁷,

Explanation:
Given (2.8)⁷ ÷ (0.7)⁷  as 2.8 ÷ 0.7 = 4 so can be written as (4)⁷ .

Question 11.
15² ÷ 25²
Answer:
\(\left(\frac{3}{5}\right)^{2}\),

Explanation:
Given 15² ÷ 25², 15² can be written as (5 X 3)² and 25²  as (5 X 5)²  ,
upon simplification we get \(\left(\frac{3}{5}\right)^{2}\).

Question 12.
7.2⁹ ÷ 2.4⁹
Answer:
(3)⁹,

Explanation:
Given (7.2)⁹ ÷ (2.4)⁹  as 7.2 ÷ 2.4 = 3, so can be written as (3)⁹.

Question 13.
(3.2x)⁹ ÷ (1.1y)⁹
Answer:
(3.2x ÷ 1.1y)⁹,

Explanation:
Given (3.2x)⁹ ÷ (1.1y)⁹ when the variable bases are different and the
powers are the same, the bases are multiplied first,
Used the powers of product property (3.2x ÷ 1.1y)⁹.

Question 14.
(-6)⁸ ÷ (-2)⁸
Answer:
(3)⁸,

Explanation:
Given (-6)⁸ ÷ (-2)⁸  as -6 can be written as (-2 x 3)⁸ so (-2)⁸ .(3)⁸ ÷ (-2)⁸
on simplification we get (3)⁸.

Question 15.
s⁵ ÷ r⁵
Answer:
\(\left(\frac{s}{r}\right)^{5}\),

Explanation:
Given s⁵ ÷ r⁵ the variable bases are different and the
powers are the same, the bases are divided first,upon simplification we get
\(\left(\frac{s}{r}\right)^{5}\).

Question 16.
(3a)⁶ ÷ (2b)⁶
Answer:
\(\left(\frac{3a}{2b}\right)^{6}\),

Explanation:
Given (3a)⁶ ÷ (2b)⁶ the variable bases are different and the
powers are the same, the bases are divided first, upon simplification we get
\(\left(\frac{3a}{2b}\right)^{6}\).

Question 17.
(h²k⁵)⁴
Answer:
h⁸k²⁰,

Explanation:
Given (h²k⁵)⁴ when the variable bases are different and the
powers are the same,Used the powers of product and
quotients of a property we get h^{2X 4} X k^5X4 = h⁸k²⁰.

Question 18.
\(\left(\frac{32 m^{6}}{4 n^{4}}\right)^{2}\)
Answer:
\(\left(\frac{8 m^{6}}{4n^{4}}\right)^{2}\),

Explanation:
Given \(\left(\frac{32 m^{6}}{4 n^{4}}\right)^{2}\) upon simplification
we get \(\left(\frac{8 m^{6}}{4n^{4}}\right)^{2}\).

Question 19.
\(\frac{9^{2} \cdot 9^{7}}{3^{5} \cdot 3^{4}}\)
Answer:
(3)⁹,

Explanation:
Given \(\frac{9^{2} \cdot 9^{7}}{3^{5} \cdot 3^{4}}\)
9 can be written as \(\frac{3 X 3 ^{2} \cdot 3 X 3^{7}}{3^{5} \cdot 3^{4}}\)
when bases are same powers are added \(\frac{3^{2+2+7+7}}{3^{5+4}}\) =
\(\frac{3^{18}}{3^{9}}\) on further simplification we get (3)^18-9 = (3)^9.

Question 20.
\(\frac{6^{5} \cdot 2^{3} \cdot 6^{4}}{12^{3}}\)
Answer:
(6)⁶,

Explanation:
Given\(\frac{6^{5} \cdot 2^{3} \cdot 6^{4}}{12^{3}}\) upon simplification
\(\frac{6^{5+4} \cdot 2^{3} }{6 X 2^{3}}\) = \(\frac{6^{9} \cdot 2^{3} }{(6 X 2)^{3}}\)=
(6)^9-3 = (6)⁶.

Question 21.
\(\frac{\left(5^{4}\right)^{2} \cdot 6^{8}}{10^{8}}\)
Answer:
(3)⁸,

Explanation:
Given \(\frac{\left(5^{4}\right)^{2} \cdot 6^{8}}{10^{8}}\) as denominator
10 can be written as 5 X 2 so \(\frac{\left(5\right)^{8} \cdot 6^{8}}{(5 X 2)^{8}}\) we get
\(\frac{6^{8}}{2^{8}}\) = \(\frac{(2 X 3)^{8}}{2^{8}}\) = (3)⁸.

Question 22.
\(\frac{\left(6^{3}\right)^{3} \cdot 4^{9}}{8^{9}}\)
Answer:
(3)⁹,

Explanation:
Given \(\frac{\left(6^{3}\right)^{3} \cdot 4^{9}}{8^{9}}\) as denominator
8 can be written as 4 X 2 so  we get
\(\frac{\left((2 X 3)^{3}\right)^{3} \cdot 4^{9}}{(4 X 2)^{9}}\) = (3)⁹.

Question 23.
\(\frac{24^{9}}{4^{3} \cdot 6^{2} \cdot 4^{6}}\)
Answer:
(6)⁷,

Explanation:
Given \(\frac{24^{9}}{4^{3} \cdot 6^{2} \cdot 4^{6}}\) as
24 in numerator can be written as 4 X 6 so

Question 24.
\(\frac{9^{12}}{\left(3^{3}\right)^{3} \cdot 3^{3}}\)
Answer:
(3)¹²,

Explanation:

Question 25.
Math Joural Charles thinks that a³ • b³ = ab⁶. Is he correct? Why?
Answer:
No, Charles is incorrect,

Explanation:
a³ • b³ ≠ ab⁶ it should be (ab)³ as when the variable bases are different and the
powers are the same, the bases are multiplied first,
Used the powers of product property we get (aX b)³ ≠ ab⁶.

Solve. Show your work.

Question 26.
At the beginning of January, Mr. Howard gives his niece $1 to start a savings account.
For each month that she can triple the amount in the account,
Mr. Howard will double the amount in the account at the end of each month.
How much does Mr. Howard’s niece have in her account at the beginning of May?
Answer:
$7,776 Mr. Howard’s niece have in her account at the beginning of May,

Explanation:
Given at the beginning of January, Mr. Howard gives his niece $1 to start a savings account.
For each month that she can triple the amount in the account,
Mr. Howard will double the amount in the account at the end of each month.
So at the beginning of January it is

therefore,$7,776 Mr. Howard’s niece have in her account at the beginning of May.

Go through the Math in Focus Grade 8 Workbook Answer Key Chapter 1 Lesson 1.6 Real-World Problems: Squares and Cubes to finish your assignments.

Math in Focus Grade 7 Course 3 A Chapter 1 Lesson 1.6 Answer Key Real-World Problems: Squares and Cubes

Math in Focus Grade 8 Chapter 1 Lesson 1.6 Guided Practice Answer Key

Solve. Show your work.

Question 1.
Find the two square roots of 169.

Answer:
Explanation:
Given to find two square roots of 169 and -169,So using prime factorization
we get for 169 it is 13 and for – 169 it is -13

Solve. Show your work.

Question 2.
Find the cube root of \(\frac{1}{729}\).

Answer:
1/9,

Explanation:

Solve. Show your work.

Question 3.
x² = 2.25
x² = 2.25

Answer:
1.5,

Explanation:

Solve. Show your work.

Question 4.

Answer:
x = (1/2),

Explanation:

Question 5.
A square field has an area of 98.01 square meters. Find the length of each side of the field.
Let the length of each side be x meters.
x² = Translate into an equation.
= Solve for x by taking the positive root of both sides.
x = m Use a calculator to find the square root.
The length of each side is meters.
Answer:
x² = 98.01 We translate into an equation:
\(\sqrt{x^{2}}\) = \(\sqrt{98.01}\) We solve for x by taking the positive square root of both sides:
x = 9.9m We use a calculator to find the square root:
The length of each side is 9.9 meters.
9.9 meters

Solve. Show your work.

Question 6.
Robin bought a crystal globe that has a volume of 1,774\(\frac{2}{3} \pi\)ir cubic centimeters. Find the radius of the crystal globe.
Let the radius of the crystal globe be r centimeters.

Answer:
The radius of the crystal globe is 29.7 cms,

Explanation:

Question 7.
A spherical waterme’on has a volume of 562.5π cubic centimeters. What is the diameter of the watermelon?

Let the radius of the watermelon be r centimeters.

Answer:
The diameter of the watermelon is 15 centimeters,

Explanation:

Math in Focus Course 3A Practice 1.6 Answer Key

Find the two square roots of each number. Round your answer to the nearest tenth when you can.

Question 1.
25
Answer:
5, or -5,

Explanation:
Given to find sqaure roots of 25
1. 5 X 5 = 25,
2. -5 X -5 = 25,
so 5, or -5.

Question 2.
64
Answer:
8 or -8,

Explanation:
Given to find sqaure roots of 64
1. 8 X 8 = 64,
2. -8 X -8 = 64,
so 8, or -8.

Question 3.
80
Answer:
8.944 or -8.944,

Explanation:
Given to find sqaure roots of 64
1. 8.944 X 8.944 = 80,
2. -8.944 X -8.944 = 80,
so 8.944, or -8.944.

Question 4.
120
Answer:
10.9544 or -10.9544,

Explanation:
Given to find sqaure roots of 64
1. 10.9554 X 10.9554 = 120,
2. -10.9554X -10.9554 = 120,
so 10.9554, or -10.5994.

Find the cube root of each number. Round your answer to the nearest tenth when you can.

Question 5.
512
Answer:
8

Explanation:

Question 6.
1,000
Answer:
10

Explanation:

Question 7.
999
Answer:
9.997,

Explanation:

Question 8.
\(\frac{64}{343}\)
Answer:
\(\frac{4}{7}\),

Explanation:
As

So Cube root of \(\frac{64}{343}\) is \(\frac{4}{7}\).

Solve each equation involving a variable that ¡s squared.
Round your answer to the nearest tenth when you can.

Question 9.
a² = 46.24
Answer:
a= 6.8,

Explanation:

Question 10.
b² = \(\frac{25}{49}\)
Answer:
b= \(\frac{5}{7}\),

Explanation:
Given b² = \(\frac{25}{49}\) as b² = \(\frac{5X5}{7X7}\),
therefore b = square root of \(\frac{5X 5}{7 X 7}\) = \(\frac{5}{7}\),

Question 11.
m² = 196
Answer:
m = 14,

Explanation:

Therefore m = 14.

Question 12.
n² = 35o
Answer:
n= 18.708,

Explanation:

therefore n = 18.708.

Solve each equation involving a variable that is cubed.
Write fractions in simplest form, and round decimal answers
to the nearest tenth.

Question 13.
x³ = 74.088
Answer:
x =4.2,

Explanation:

the cube root of 74.088 is 4.2.

Question 14.
x³ = \(\frac{216}{729}\)
Answer:
x = \(\frac{6}{9}\),

Explanation:

the cube root of x³ = \(\frac{216}{729}\) so x = \(\frac{6}{9}\).

Question 15.
x³ = 1,728
Answer:
x= 12,

Explanation:

Question 16.
x³ = 2,500
Answer:
x= 13.572,

Explanation:

the cube root x = 13.572088083.

Solve. Show your work. Round to the nearest tenth.

Question 17.
The volume of a spherical tank is 790.272π cubic feet.
What is the diameter of the container?

Answer:
The diameter of the container is 16.8 ft,

Explanation:
Given the volume of a spherical tank is 790.272π cubic feet.
so radius is 4/3 pi r³ = 790.272π,
r³ = (790.272 X 3)/4 = 592.704,
r= 8.4,

diameter of the container is 2r = 2 X 8.4 = 16.8 ft.

Question 18.
An orchard planted on a square plot of land has 3,136 apple trees.
If each tree requires an area of 4 square meters to grow,
find the length of each side of the plot of land.

Answer:
The length of each side of plot is 112 meters,

Explanation:
Given an orchard planted on a square plot of land has 3,136 apple trees.
If each tree requires an area of 4 square meters to grow,
the length of each side of the plot of land of square root of 3136 X 4
sqaure root of 12544 = 112 meters.

Question 19.
Mr. Berman deposited $2,500 in a savings account. Three years later there
was $2,812.16 in the savings account. Use the formula A = P(1 + r)ⁿ to find
the rate of interest, r percent, that he was paid. A represents the final amount of the
investment, P is the original principal, and n is the number of years it was invested.
Answer:
Rate of intreset is 5.78%,

Explanation:
Given Mr. Berman deposited $2,500 in a savings account. Three years later there
was $2,812.16 in the savings account. Using the formula A = P(1 + r)ⁿ to find
the rate of interest, r percent, that he was paid. A represents the final amount of the
investment, P is the original principal, and n is the number of years it was invested,
$2812.16 = $2,500(1 + r)³
$2812.16 – $2,500 = (1 + r)³
(1 + r)³ = $312.16
1 + r = cube root of $312.16,

1 + r = 6.7835,
therefore r = 6.7835 – 1 = 5.7835.

Brain @ Work

Question 1.
Evaluate \(\frac{4^{3} \cdot 10^{4}}{5^{2}}\) without using a calculator.
Answer:
2¹⁰ X 5²,

Explanation:

Question 2.
Find the values of x and y that make the equation \(\frac{81 x^{4} \cdot 16 y^{4}}{\left[(2 y)^{2}\right]^{2}}\) = 1,296 true,
Answer:
X = 1.414 and y = 1,

Explanation:

Question 3.
Use each of the numbers 1 to 9 exactly once to fill in the blanks.

Answer:
3¹¹,

Explanation:
Used each of the numbers 1 to 9 exactly once to fill the blanks

Question 4.
Jeremy wants to measure the radius of a marble. He uses a tank and
filled the tank with 360 identical marbles shown below, If the volume of the
tank is 9,720 cubic inches, find the radius of each marble.

Answer:
Radius is 3 inches,

Explanation:
Given Jeremy wants to measure the radius of a marble. He uses a tank and
filled the tank with 360 identical marbles shown below, If the volume of the
tank is 9,720 cubic inches, volume of each cube is 9,720 ÷ 360 = 27,
radius is cube root of 27 = 3 inches.

Go through the Math in Focus Grade 8 Workbook Answer Key Chapter 1 Lesson 1.5 Zero and Negative Exponents to finish your assignments.

Math in Focus Grade 7 Course 3 A Chapter 1 Lesson 1.5 Answer Key Zero and Negative Exponents

Math in Focus Grade 8 Chapter 1 Lesson 1.5 Guided Practice Answer Key

Hands-On Activity

Understand the zero exponent

Work individually.

Step 1.
Use the power of a quotient property to simplify each expression. Write the quotient as an exponent.

Step 2.
In factored form, the quotient \(\frac{3^{5}}{3^{5}}\) is \(\frac{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}\). If you divide out all the common factors in the numerator and denominator, what is the value of \(\frac{3^{5}}{3^{5}}\)?

Step 3.
Based on your findings, what can you conclude about the value of 3°?

Step 4.
Make a prediction about the value of any nonzero number raised to the zero power. Then, use a calculator to check your prediction for several numbers. For example, to raise the number —2 to the zero power, you can enter these keystrokes:

Does your prediction hold true?
Answer:
Step 1:

Wrote the quotient as an exponent of 3° used exponent \(\frac{3^{5}}{3^{5}}\),
Step 2:
Value is 1,
Step 3:
The value of 3° is 1.
Step 4:
Yes,

Explanation:
Step 1:
Used the power of a quotient property to simplify each expression.
Wrote the quotient as an exponent of 3° used exponent \(\frac{3^{5}}{3^{5}}\),
Step2:
In factored form, the quotient \(\frac{3^{5}}{3^{5}}\) is \(\frac{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}\). If you divide out all the common factors in the numerator and denominator the value of \(\frac{3^{5}}{3^{5}}\) is 1 ,
Step 3 :
Any number to power 0 the value is 1, so the value of 3° is 1,
Step 4:
Makeing a prediction about the value of any nonzero number raised to the zero power.
Then, using a calculator to check your prediction for several numbers.
For example, to raise the number —2 to the zero power, you can enter these keystrokes:

Yes my predictions hold true as
Any number to power 0 the value is 1.

Simplify each expression and evaluate where applicable.

Question 1.

Answer:
(0.4)^-2,

Explanation:

Question 2.

Answer:
1,

Explanation:

Question 3.
\(\frac{t^{0} \cdot t^{7}}{t^{5}}\)
Alternatively, you can also apply the product of powers property to the expression t° ∙ t⁷ to get t⁰⁺⁷ = t⁷.
Answer:
t²,

Explanation:
Given\(\frac{t^{0} \cdot t^{7}}{t^{5}}\) as t⁰ is equal to 1  so
we get 1 X t^7-5= t².

Hands-On Activity

Work individually.

Step 1.
Use the quotient of powers property to simplify each expression. Write the quotient in exponential notation.

Answer:

Explanation:
Wrote the quotient in exponential notation above,
As (4)⁵÷ (4)⁶ = (4)^5-6= (4)^-1 used negative exponent.

Step 2.
In factored form, the quotient \(\frac{4^{5}}{4^{6}}\) is \(\frac{4 \cdot 4 \cdot 4 \cdot 4 \cdot 4}{4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4}\). If you divide out all the common factors in the numerator and denominator, what is the value of
\(\frac{4^{5}}{4^{6}}\)?

Answer:
The value is (4)^-1,

Explanantion:
(4)⁵÷ (4)⁶ = (4)^5-6= (4)^-1 used negative exponent.

Step 3.
Repeat Step 2 for \(\frac{4^{5}}{4^{7}}\). What is the value of \(\frac{4^{5}}{4^{7}}\)?

Answer:
The value is (4)^-2,

Explanantion:
(4)⁵÷ (4)⁷ = (4)^5-7= (4)^-2.

Math Journal
Suppose a represents any nonzero number. How would you write a^-3 using a positive exponent?

Simplify each expression and evaluate where applicable.
Answer:
1 ÷ a³ ,

Explanation:
Given  a^-3 using a positive exponent for any nonzero real
number a and any integer 3, a^-3 = 1 ÷ a³ ,a ≠ 0.

Question 4.

Answer:
(2.5)^-3,

Explanantion:

Question 5.

Answer:
(-6)^-1,

Explanation:

Simplify each expression. Write your answer using a positive exponent.

Question 6.
14a^-5 ÷ (7a • 2a^-4)
14a^-5 ÷ (7a • 2a^-4) = Write the expression as a .
=
Answer:
1,

Explanation:

Usually you should write your answer using a positive exponent
unless asked to use a negative exponent.

Math in Focus Course 3A Practice 1.5 Answer Key

Simplify each expression and evaluate.

Question 1.
8³ • 8°
Answer:
8³,

Explanation:
Given 8³ • 8° any number to power 0 the value is 1, so the value of 8° is 1,
so 8³ • 8° = 8³ • 1 = 8³.

Question 2.
5⁴ • (-5)°
Answer:
5⁴,

Explanation:
Given 5⁴ • (-5)° any number to power 0 the value is 1, so the value of (-5)° is 1,
so 5⁴ • (-5)° = 5⁴ • 1 = 5⁴.

Question 3.
\(\left(\frac{1}{3}\right)^{4} \cdot\left(\frac{1}{3}\right)^{0}\)
Answer:
\(\left(\frac{1}{3}\right)^{4}\),

Explanation:
Given \(\left(\frac{1}{3}\right)^{4} \cdot\left(\frac{1}{3}\right)^{0}\) as
\(\left(\frac{1}{3}\right)^{0}\) = 1 therefore \(\left(\frac{1}{3}\right)^{4}\) X 1 =
\(\left(\frac{1}{3}\right)^{4}\).

Question 4.
7 • 10³ + 4² • 10² + 5 • 10°
Answer:
8,605,

Explanation:
Given 7 • 10³ + 4² • 10² + 5 • 10° solving as 7 X 10 X 10 X 10 + 4 X 4 X 10 X 10 + 5 X 1 =
7 X 1,000 + 16 X 100 + 5 = 7,000 + 1,600 + 5 = 8,605.

Question 5.
(2.3) • 10² + 5 • 10¹ + 1 • 10°
Answer:
281,

Explanation:
Given (2.3) • 10² + 5 • 10¹ + 1 • 10° solving as (2.3) X 10 X 10 + 5 X 10 + 1 X 1 =
230 + 50 + 1 =281.

Question 6.
\(\frac{7^{4} \cdot 7^{5}}{7^{9}}\)
Answer:
1,

Explanation:
Given \(\frac{7^{4} \cdot 7^{5}}{7^{9}}\) solving
\(\frac{7^{4+5}}{7^{9}}\) = \(\frac{7^{9}}{7^{9}}\)=
(7)^9-9 =(7)⁰ = 1.

Question 7.
(9^-3)° • 5²
Answer:
5²,

Explanation:
Given (9^-3)° • 5² any number to power 0 the value is 1,
so the value of (9^-3)°  is 1, therefore 1 X 5² = 5².

Question 8.
\(\frac{\left(6^{-3}\right)^{-2} \cdot 8^{6}}{48^{6}}\)
Answer:
1

Explanation:

Simplify each expression. Write your answer using a negative exponent.

Question 9.
7³ • 7^-4
Answer:
7^-1,

Explanation:
Given 7³ • 7^-4  using products of powers property we get
7^3-4 = 7^-1.

Question 10.
\(\frac{(-5)^{-2}}{(-5)^{3}}\)
Answer:
\(\frac{(-5)^{-2}}{(-5)^{3}}\) simplify
= (-5)^-1

Question 11.

Answer:

Question 12.
\(\left(\frac{2}{5}\right)^{-4} \cdot\left(\frac{2}{5}\right)^{-1} \div\left(\frac{2}{5}\right)^{-3}\)
Answer:
\(\left(\frac{2}{5}\right)^{-2}\),

Explanation:

Question 13.
\(\frac{x^{0}}{x^{2} \cdot x^{3}}\)
Answer:
x ^-5,

Explanation:

Question 14.
\(\frac{4 h^{-5} \cdot 6 h^{-2}}{3 h^{-3}}\)
Answer:
8(h)^-4,

Explanation:

Simplify each expression. Write your answer ustng a positive exponent.

Question 15.
1.2° ÷ 1.8²
Answer:
0.308,

Explanation:
Given 1.2° ÷ 1.8² any number to power 0 the value is 1,
so the value of (1.2)° is 1, therefore 1÷1.8² = 1 ÷ 3.24 = 0.308.

Question 16.
5.2^-3 ÷ 2.6^-3
Answer:
0.125,

Explanation:
Given 5.2^-3 ÷ 2.6^-3  we get  2.6³ ÷ 5.2³ upon solving
(2.6 X 2.6 X 2.6) ÷ (5.2 X 5.2 X 5.2) = 0.125.

Question 17.
\(\frac{(-3)^{-4}}{(-3)^{2}}\)
Answer:
\(\frac{(1)}{(-3)^{6}}\),

Explanation:
Given

Question 18.
\(\left(\frac{5}{6}\right)^{-4} \cdot\left(\frac{5}{6}\right)^{-2} \div\left(\frac{5}{6}\right)^{-3}\)
Answer:
\(\frac{(1)}{\frac{5}{6}^{3}}\),

Explanation:

Question 19.
\(\frac{9 k^{-1} \cdot 2 k^{-3}}{27 k^{-6}}\)
Answer:
2k² ÷ 3,

Explanation:

Question 20.
\(\frac{c^{-4} \cdot c^{12}}{c^{-7}}\)
Answer:
(c)¹⁵,

Explanation:

Evaluate each numeric expression.

Question 21.
\(\frac{7^{-2} \cdot 7^{0}}{8^{3} \cdot 8^{-5}}\)
Answer:
(8/7)²,

Explanation:

Question 22.
\(\frac{\left(7^{-2}\right)^{2} \cdot 9^{-4}}{21^{-4}}\)
Answer:
(3)^-4,

Explanation:

Question 23.
\(\frac{10^{0}}{2^{-2} \cdot\left(5^{-1}\right)^{2}}\)
Answer:
100,

Explanation:

Question 24.
\(\frac{\left(3^{6}\right)^{-2}}{6^{-9}\left(-2^{10}\right)}\)
Answer:
(-1/54),

Explanation:

Simplify each algebraic expression.

Question 25.
\(\left(\frac{7 m^{3}}{-49 n^{0}}\right)^{-1}\)
Answer:
(-7)m^-3,

Explanation:

Question 26.
\(\frac{8 r^{2} s}{4 s^{-3} r^{4}}\)
Answer:
2r^-2s⁴,

Explanation:

Go through the Math in Focus Grade 8 Workbook Answer Key Chapter 1 Review Test to finish your assignments.

Math in Focus Grade 8 Course 3 A Chapter 1 Review Test Answer Key

Concepts and Skills

Identify the base and exponent in each expression.

Question 1.
\(\left(-\frac{1}{5}\right)^{-3}\)
Answer:
base is \(\left(-\frac{1}{5}\right)\) and exponent is -3,

Explanation:
Given \(\left(-\frac{1}{5}\right)^{-3}\) so base is
base value is \(\left(-\frac{1}{5}\right)\) and exponent value is -3,

Question 2.
-0.92⁴
Answer:
base is (-0.92) and exponent is 4,

Explanation:
Given -0.92⁴ here the base is -0.92 and exponent is 4.

Tell whether each statement is correct. If it is incorrect, state the reason.

Question 3.
-0.7³ = -0.7 • 0.7 • 0.7
Answer:
Statement is incorrect,

Explanation:
Given -0.7³ = -0.7 • 0.7 • 0.7 which is incorrect the correct statement is
-0.7³ = -0.7 • -0.7 • -0.7.

Question 4.
5^-4 = (-5) • (-5) • (-5) • (-5)
Answer:
Statement is correct,

Explanation:
Given statement is
5^-4 = (-5) • (-5) • (-5) • (-5) is correct.

Write in exponential notation.

Question 5.
2 • 2 • 2 • 2
Answer:
2⁴,

Explanation:
Given 2 • 2 • 2 • 2 here base is 2 and exponent is 4 therefore
the exponential notation is 2⁴.

Question 6.
4.8 • 4.8
Answer:
4.8²,

Explanation:
Given 4.8 • 4.8 here base is 4.8 and exponent is 2 therefore
the exponential notation is 4.8².

Question 7.
\(\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}\)
Answer:
\(\frac{1}{2}\),

Explanation:
Given \(\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}\) here base is \(\frac{1}{2}\) and exponent is 3 therefore the exponential notation is \(\left(\frac{1}{2}\right)^{3}\).

Question 8.
c • c • c • c • c • c
Answer:
c⁶,

Explanation:
Given c • c • c • c • c • c here base is c and exponent is 6 therefore
the exponential notation is c⁶.

Question 9.
\(\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}\)
Answer:
\(\frac{1}{2}\),

Explanation:
Given \(\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}\) here base is \(\frac{1}{2}\) and exponent is 3 therefore the exponential notation is \(\left(\frac{1}{2}\right)^{3}\).

Question 10.
(-1.2)(-1.2)(-1.2)(-1.2)
Answer:
(-1.2)⁴,

Explanation:
Given (-1.2)(-1.2)(-1.2)(-1.2)  here base is (-1.2) and exponent is 4 therefore
the exponential notation is (-1.2)⁴.

Write the prime factorization of each number in exponential notation.

Question 11.
3,780
Answer:
Prime factorization of 3,780 = 2² X 3³ X 5 X 7,

Explanation:

Question 12.
27,720
Answer:
Prime factorization of 27,720 = 2³ X 3² X 5 X 7 X 11,

Explanation:

Expand and evaluate each expressions.

Question 13.
(-6)²
Answer:
36,

Explanation:
Given (-6)² so upon expanding we get (-6) X (-6) = 36.

Question 14.
1.1²
Answer:
1.21,

Explanation:
Given (1.1)² so upon expanding we get (1.1) X (1.1) = 1.21.

Question 15.
10⁵
Answer:
100,000

Explanation:
Given 10⁵ so upon expanding we get 10 X 10 X 10 X 10 X 10 = 100,000.

Question 16.
\(\left(\frac{2}{3}\right)^{3}\)
Answer:
\((\frac{8}{27}\),

Explanation:
Given\(\left (\frac{2}{3}\right)^{3}\) so upon expanding we get
\((\frac{2}{3})\) X \((\frac{2}{3})\) X \((\frac{2}{3})\) =
\((\frac{2X 2 X 2}{3 X 3 X 3})\) = \((\frac{8}{27})\).

Simplify each expression. Write your answer using a positive exponent.

Question 17.
(-3)^-1 . (-3)°
Answer:
\(\frac{1}{-3}\),

Explanation:
Given (-3)^-1 . (-3)° as (-3)°  is 1 and (-3)^-1 value in positive exponent is
\(\frac{1}{-3}\).

Question 18.
\(\left(\frac{5}{6}\right)^{4} \cdot\left(\frac{5}{6}\right)^{3}\)
Answer:

Question 19.
5m³n⁴ . 4m⁵n²
Answer:
20m⁸n⁶,

Explanation:
Given 5m³n⁴ . 4m⁵n² on solving 5 X 4 X m³⁺⁵ X n⁴⁺² = 20m⁸n⁶.

Question 20.
\(\left(\frac{7}{8}\right) \div\left(\frac{7}{8}\right)^{3}\)
Answer:

Question 21.
(-3)^-1 • (-3)°
Answer:
\(\frac{1}{-3}\),

Explanation:
Given (-3)^-1 . (-3)° as (-3)°  is 1 and (-3)^-1 value in positive exponent is
\(\frac{1}{-3}\).

Question 22.
x⁸z⁵ ÷ x³z⁹
Answer:
x⁵z^-4,

Explanation:
Given x⁸z⁵ ÷ x³z⁹ using division of product property
x^8-3z^5-9 we get x⁵z^-4.

Question 23.
25p⁶q⁹ ÷ 45p⁸q⁴
Answer:
5p^-2 q⁵ ÷ 9,

Explanation:
Given 25p⁶q⁹ ÷ 45p⁸q⁴
= 5p^6-8 q^9-4 ÷ 9,
= 5p^-2 q⁵ ÷ 9,

Question 24.
\(\left[\left(\frac{2}{3}\right)^{2} \cdot\left(\frac{2}{3}\right)^{-1}\right]^{3}\)
Answer:

Question 25.
40c⁵d³ ÷ 10c⁹d²
Answer:
4c^-4d,

Explanation:
Given 40c⁵d³ ÷ 10c⁹d² =4 X 10 X c^5-9 X d^3-2 ÷ 10 = 4c^-4d.

Question 26.

\(\left(\frac{72 b^{-1}}{32 c^{-1}}\right)^{-2}\)
Answer:

Question 27.
\(\frac{\left(9^{-2}\right)^{-2} \cdot 2^{2}}{9^{2}}\)
Answer:
9² X 2²,

Explanation:
Given

Question 28.
\(\frac{6^{8} \cdot 56^{-3}}{6^{5} \cdot 7^{-3}}\)
Answer:
6³ X 8^-3,

Explanation:

Question 29.
\(\frac{42^{-1}}{\left(2^{0}\right)^{12} \cdot 21^{-1}}\)
Answer:
2^-1,

Explanation:

Question 30.
\(\frac{\left(3^{5} \cdot 3^{4}\right)^{2}}{\left(3^{3}\right)^{6}}\)
Answer:
1,

Explanation:

Solve each equation involving a variable that is squared.

Question 31.
r² = 256
Answer:
r = 16,

Explanation:

Question 32.
c² = \(\frac{121}{169}\)
Answer:
c = \(\frac{11}{13}\),

Explanation:
Given c² = \(\frac{121}{169}\)= \(\frac{11 X 11}{13 X 13}\),
c is sqaure root of \(\frac{11 X 11}{13 X 13}\) = \(\frac{11}{13}\).

Solve each equation involving a variable that is cubed.

Question 33.
x^{3 = 32.768
Answer:}
x = 3.2,

Explanation:

Question 34.
c³ = –\(\frac{27}{343}\)
Answer:
c= – \(\frac{3}{7}\),

Explanation:
Given c³ = –\(\frac{27}{343}\) can be written as
c³ = –\(\frac{3 X 3 X 3}{7 X 7 X 7}\) therefore c is cube root of
–\(\frac{3 X 3 X 3}{7 X 7 X 7}\) = – \(\frac{3}{7}\)

Question 35.
The expanded form of a number is 5 • 10¹ + 8 • 10° + 1 • 10^-1 + 9 • 10^-2. What is this number in standard form?
Answer:
The number in standard form is 58.19,

Explanation:
Given 5 • 10¹ + 8 • 10° + 1 • 10^-1 + 9 • 10^-2= 5 X 10 + 8 X 1 + 1 X 0.1 + 9 X 0.01 =
50 + 8 +0.1 + 0.09 = 58.19.

Question 36.
The pattern of triangles shown is called the Sierpinski’s gasket.
a) Find a pattern in the number of shaded triangles.
Answer:
(3/4)ⁿ,

Explanation:
The concept of the Sierpinski triangle is very simple:
Take a triangle. Create four triangles out of that one by connecting the centres of each side.
So basically each triangle cuts into four and each again cuts four and so on.
So, there are an infinite number of triangles pattern is (3/4)ⁿ.

b) How many shaded triangles will be in the fifth diagram of this pattern?
Write an exponential expression for this number.

Answer:
Shaded triangles will be in the fifth diagram of this pattern are 243.
Explanation:
We have pattern(3/4)ⁿ here in first shape we have
3 shaded triangles out of 4, 5th pattern will have (3/4)⁵ so
shaded triangles will be in the fifth diagram of this pattern
3 X 3 X 3 X 3 X 3 = 243.

Problem Solving

Solve. Show your work. Round your answer to the nearest tenth.

Question 37.
The Reina Sofia Museum in Spain has a glass elevator.
The floor of the elevator shaft is a square with an area of about 42.25 square feet.
Find the length of a side of the floor.
Answer:
The length of a side of the floor is 6.5 feet,

Explanation:
Given the Reina Sofia Museum in Spain has a glass elevator.
The floor of the elevator shaft is a square with an area of about 42.25 square feet.
The length of a side of the floor is 6.5 feet.

Question 38.
Earth’s volume is approximately 1,083,210,000,000 cubic kilometers.
What is the diameter of the Earth in kilometers?

Answer:
The diameter of the earth is 12,562 km ,

Explanation:
Given Earth’s volume is approximately 1,083,210,000,000 cubic kilometers.,
4/3pir³ = 1,08321 X 10^{7 ,}
we get r = 6281 kms,
therefore the diameter of the earth is 2 X 6281 km =  12,562 km.

Question 39.
Koch’s snowflake is a pattern that starts with an equilateral triangle.
Then, in successive images, each line segment is replaced by 4 line segments, as shown below.
How many line segments make up the seventh image of this pattern?
Write this number in exponential form and evaluate.

Answer:
3 X 4⁶,

Explanation:
Given Koch’s snowflake is a pattern that starts with an equilateral triangle.
Then, in successive images, each line segment is replaced by 4 line segments, as shown above.
we can express it in sequence of numbers
a1=3,
a2 = 3 X 4 = 12,
a3 = 3 X 4 X 4 = 48,
an = a^(n-1) X 4 = 4 a^(n-1),(geometric progression)
an = 3 x 4^(n-1),
n= 7
a7 = 3 X 4⁶.