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.

6^{4} • 6^{3} =

6^{4} • 6^{3} = Use the of powers property.

= Simplify.

Answer:

6^{7},

Explanation:

Given 6^{4} • 6^{3} 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 6^{4} • 6^{3 }= 6^{4 + 3 }= 6^{7}.

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.

p^{3} • p^{6}

p^{3} • p^{6} = Use the of powers property.

= Simplify.

Answer:

p^{3} • p^{6} = p^{3 + 6} = p^{9},

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.

pq^{3} • p^{5}q^{2}

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.

4s^{4}t^{3} • 5s^{4}t^{6}

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.

10^{8} ÷ 10^{5}

10^{8} ÷ 10^{5} = Use the of powers property.

= Simplify.

Answer:

10^{8} ÷ 10^{5} = 10^{3},

Explanation:

Given to find 10^{8} ÷ 10^{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 10^{8} ÷ 10^{5} = 10^{8-5}= 10^{3}.

2.7

^{9}÷ 2.7

^{6}=

= simplify

Answer:

2.7

^{9}÷ 2.7

^{6}= 2.7

^{9-6}= 2.7

^{3},

Explanation:

Given to find 2.7^{9} ÷ 2.7^{6} 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 2.7^{9} ÷ 2.7^{6} = 2.7^{9-6}= 2.7^{3}.

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.

q^{7} ÷ q^{2}

q^{7} ÷ q^{2} =

= Simplify

Answer:

\({q}^{5}\),

Explanation:

Given to find q^{7} ÷ q^{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 q^{7} ÷ q^{2} = q^{7-2}= q^{5}.

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.^{2}3 • 7.2^{4}

Answer:

7.2^{3} . 7.2^{4} = 7.2^{3+4} = 7.2^{7},

Explanation:

Given to find 7.2^{3} . 7.2^{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, So 7.2^{3} **.** 7.2^{4} = 7.2^{3+4}= 7.2^{7}.

Question 3.

10^{5} • 10^{4}

Answer:

10^{9}

Explanation:

Given 10^{5} • 10^{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, So 10^{5} • 10^{4} = 10^{5+4} = 10^{9}.

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 • p^{8}

Answer:

p^{1} • p^{8} = p^{1 + 8} = p^{9},

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.

q^{8} ÷ q

Answer:

\({q}^{7}\),

Explanation:

Given to find q^{8} ÷ q^{1} 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 q^{8} ÷ q^{1} = q^{8-1}= q^{7}.

Question 7.

xy^{2} • x^{4}y^{3}

Answer:

\({x}^{5}\) • \({y}^{5}\),

Explanation:

Given xy^{2} • x^{4}y^{3}

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.

2x^{2}y^{4} • 5x^{5}y

Answer:

10 X \({x}^{7}\) • \({y}^{5}\),

Explanation:

Given 2x^{2}y^{4} • 5x^{5}y 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.

2.5x^{3}y^{6} • 3x^{2}y^{4}

Answer:

7.5 X \({x}^{5}\) • \({y}^{10}\),

Explanation:

Given 2.5 X x^{3}y^{6} • 5x^{2}y^{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 (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.

2^{10} ÷ 2^{5}

Answer:

2^{5},

Explanation:

Given to find 2^{10} ÷ 2^{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 2^{10} ÷ 2^{5} = 2^{10-5}= 2^{5}.

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.

63y^{3}z^{5} ÷ 9

Answer:

7y^{3}z^{5},

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.

h^{2}k^{5} ÷ hk^{4}

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.

64a^{8}b^{5} ÷ 4a^{3}b^{2}

Answer:

16a^{5}b^{3},

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. a^{5}b^{3} .

Question 16.

\(\frac{5^{9} \cdot 5^{7} \cdot 5^{8}}{5^{3} \cdot 5^{2} \cdot 5}\)

Answer:

5^{18},

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:

2x^{4}z^{3},

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:

9bc^{5},

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 = 4x^{1+1 } = 4x^{2} units,

height as 2 X x = 2x units and length as 3x X 3 x = 9x^{1+1 }= 9x^{2 }units,

So the volume of the rectangular prism using exponential notation is

w X h X l = 4x^{2} X 2x X 9x^{2} = 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:

12x^{2} 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}.