Go through the Math in Focus Grade 6 Workbook Answer Key Chapter 1 Lesson 1.5 Cubes and Cube Roots to finish your assignments.

## Math in Focus Grade 6 Course 1 A Chapter 1 Lesson 1.5 Answer Key Cubes and Cube Roots

### Math in Focus Grade 6 Chapter 1 Lesson 1.5 Guided Practice Answer Key

**Find a cube of a whole number.**

a) A cube has edges 2 centimeters long. Find its volume.

Volume of cube = 2 × 2 × 2 = 8 cm^{3}

2 × 2 × 2 is called the cube of 2.

You can write 2 × 2 × 2 as 2^{3}.

So, 2^{3} = 8.

The number 3 in 2^{3} is the exponent. The number 2 is the base.

The cube of a whole number is called a perfect cube.

Since 8 = 2 × 2 × 2, 8 is a perfect cube.

b) Find the cube of 7.

7^{3} = 7 × 7 × 7

= 343

The cube of 7 is 343.

I can relate this to finding the volume of a cube with edges of length 7 units.

**Find the cube of each number.**

Question 1.

5

Answer:

125

Explanation:

I can relate this to finding the volume of a cube with edges of length 5 units.

^{5³ = 5 x 5 x 5}

= 125

The cube of 5 is 125.

Question 2.

6

Answer:

216

Explanation:

I can relate this to finding the volume of a cube with edges of length 7 units.

^{7³ = 7 x 7 x 7}

= 216

The cube of 6 is 216.

Question 3.

9

Answer:

729

Explanation:

I can relate this to finding the volume of a cube with edges of length 9 units.

^{9³ = 9 x 9 x 9}

= 729

The cube of 5 is 729.

**Find a cube root of a perfect cube.**

a) A cube has a volume of 27 cubic meters. Find the length of each edge of the cube.

You know that

Volume of cube = edge × edge × edge.

To find the length of the edge of the cube, you need to find a number whose cube is 27.

You know that 3 × 3 × 3 = 27.

So, the length of each edge of the cube is 3 meters. 3 is called the cube root of 27. This can be written as

\(\sqrt[3]{27}\) = 3

Finding the cube root of a number is the inverse of finding the cube of a number.

b) Find the cube root of 64.

By prime factorization,

I can relate this to finding the length of an edge of a cube, when I know it has a volume of 64 units^{3}.

**Find the cube root of each number.**

Question 4.

216

Answer:

6

Explanation:

Finding the cube root of a number is the inverse of finding the cube of a number

prime factorization of 216 = 2 x 2 x 2 x 3 x 3 x 3

= (2.3)³

=6³

so, cube root of 216 is 6.

Question 5.

343

Answer:

7

Explanation:

Finding the cube root of a number is the inverse of finding the cube of a number

prime factorization of 343 = 7 x 7 x 7

=7³

so, cube root of 343 is 7.

Question 6.

1,000

Answer:

10

Explanation:

Finding the cube root of a number is the inverse of finding the cube of a number

prime factorization of 1000 = 2 x 2 x 2 x 5 x 5 x 5

= (2.5)³

=10³

so, cube root of 1000 is 10.

**Complete.**

Question 7.

Find the values of 5^{2} + 5^{3} and 5 • 5 • 5 • 5 • 5.

Answer:

**Find the value of each of the following.**

Question 8.

6^{3} + 4^{2}

Answer:

232

Explanation:

6³ = 6 x 6 x 6= 216

4² = 4 x 4 = 16

6³ + 4² = 216 + 16 = 232

6³ + 4² = 232

Question 9.

7^{3} – 4^{3}

Answer:

279

Explanation:

7³ = 7 x 7 x 7 = 343

4³ = 4 x 4 x 4 = 64

7³ – 4³ = 343 – 64 = 279

7³ – 4³ = 279

Question 10.

3^{2} × 5^{3} + 9^{2}

Answer:

215

Explanation:

3² = 3 x 3 = 9

5³ = 5 x 5 x 5 = 125

9² = 9 x 9 = 81

3² x 5³ x 9² = 9+125+81 = 215.

3² x 5³ x 9² = 215.

Question 11.

8^{3} ÷ 4^{2} – 5^{2}

Answer:

7

Explanation:

8³ = 8 x 8 x 8 = 512

4² = 4 x 4 = 16

5² = 5 x 5 = 25

8³ ÷ 4² – 5² = 512 ÷ 16 – (25)

= 32 – 25

= 7

8³ ÷ 4² – 5² = 7

Question 12.

7^{2} + 6^{3} ÷ 2^{3}

Answer:

76

Explanation:

7^{2} = 7 x 7 = 49

6³ = 6 x 6 x 6 = 216

2³ = 2 x 2 x 2 = 8

7^{2} + 6^{3} ÷ 2³ = 49 + (216 ÷ 8)

= 49 + 27

= 76

7^{2} + 6^{3} ÷ 2³ = 76

Question 13.

9^{3} – 4^{2} × 3^{3}

Answer:

297

Explanation:

9³ = 9 x 9 x 9 = 729

4² = 4 x 4 = 16

3³ = 3 x 3 x 3 = 27

9³ – 4² + 3³ = 729 – (16 x 27)

= 729 – 432

= 279

9³ – 4² + 3³ = 279.

### Math in Focus Course 1A Practice 1.5 Answer Key

**Find the cube of each number.**

Question 1.

8

Answer:

512

Explanation:

I can relate this to finding the volume of a cube with edges of length 8 units.

^{8³ = 8 x 8 x 8
}= 64 x 8

= 512

The cube of 8 is 512.

Question 2.

3

Answer:

27

Explanation:

I can relate this to finding the volume of a cube with edges of length 3 units.

^{3³ = 3 x 3 x 3
}9 x 3

= 27

The cube of 3 is 27.

Question 3.

10

Answer:

1000

Explanation:

I can relate this to finding the volume of a cube with edges of length 10 units.

^{10³ = 10 x 10 x 10
}=100 x 10

= 1000

The cube of 10 is 1000.

**Find the cube root of each number.**

Question 4.

125

Answer:

5

Explanation:

Finding the cube root of a number is the inverse of finding the cube of a number

prime factorization of 125 = 5 x 5 x 5

=5³

so, cube root of 125 is 5.

Question 5.

512

Answer:

8

Explanation:

Finding the cube root of a number is the inverse of finding the cube of a number

prime factorization of 512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

= (2.2.2)³

=8³

so, cube root of 512 is 8.

Question 6.

729

Answer:

9

Explanation:

Finding the cube root of a number is the inverse of finding the cube of a number

prime factorization of 729 = 3 x 3 x 3 x 3 x 3 x 3

= (3.3)³

=9³

so, cube root of 729 is 9.

**Solve.**

Question 7.

List the perfect cubes that are between 100 and 600.

Answer:

125, 216, 343 and 512.

Explanation:

125, 216, 343 and 512 are the perfect cubes between 100 and 600.

Question 8.

**Math Journal** Find the value of each expression. Then describe any patterns you see.

a) 2^{2} – 1^{2}

Answer:

3

Explanation:

2² = 2 x 2 = 4

1² = 1 x 1 = 1

2² – 1² = 4 – 1

= 3

2² – 1² = 3.

b) 3^{2} – 2^{2}

Answer:

5

Explanation:

3² = 3 x 3 = 9

2² = 2 x 2 = 4

3² – 2² = 9 – 4

= 5

3² – 2² = 5.

c) 4^{2} – 3^{2}

Answer:

7

Explanation:

4² = 4 x 4 = 16

3² = 3 x 3 = 9

4² – 3² = 16 – 9

= 7

4² – 3² = 7.

d) 5^{2} – 4^{2}

Answer:

9

Explanation:

5² = 5 x 5 = 25

4² = 4 x 4 = 16

5² – 4² = 25 – 16

= 9

5² – 4² = 9.

Question 9.

Find two consecutive numbers whose squares differ by 17.

Answer:

8 and 9

Explanation:

The square of 8 is 64

The square of 9 is 81

81 – 64 = 17

So, 8 and 9 are two consecutive numbers whose squares differ by 17.

**Find the value of each of the following.**

Question 10.

8^{3} + 5^{2}

Answer:

537

Explanation:

8³ = 8 x 8 x 8 = 512

5² = 5 x 5 = 25

8³ + 5² = 512 + 25

= 537

8³ + 5² = 537.

Question 11.

10^{3} – 6^{2}

Answer:

964

Explanation:

10³ = 10 x 10 x 10 = 1000

6² = 6 x 6 = 36

10³ – 6² = 1000 – 36

= 964

10³ – 6² = 964.

Question 12.

3^{3} × 9^{2}

Answer:

2187

Explanation:

3³ = 3 x 3 x 3 = 27

9² = 9 x 9 = 81

3³ x 9² = 27 x 81

= 2187

3³ x 9² = 2187

Question 13.

9^{3} – 5^{2} + 6^{2}

Answer:

740

Explanation:

9³ = 9 x 9 x 9 = 729

5² = 5 x 5 = 25

6² = 6 x 6 = 36

9³ – 5² + 6² = 729 – 25 + 36

= 740

9³ – 5² + 6² = 740.

Question 14.

7^{2} + 8^{3} – 4^{2}

Answer:

545

Explanation:

7² = 7 x 7 = 49

8³ = 8 x 8 x 8 = 512

4² = 4 x 4 = 16

7^{2} + 8^{3} – 4² = 49 + 512 – 16

= 545

7^{2} + 8^{3} – 4² = 545

Question 15.

9^{3} – 5^{2} + 6^{2}

Answer:

740

Explanation:

9³ = 9 x 9 x 9 = 729

5² = 5 x 5 = 25

6² = 6 x 6 = 36

9^{3} – 5^{2} + 6² = 729 – 25 + 36

= 740

9^{3} – 5^{2} + 6² = 740

Question 16.

8^{3} × 5^{3} ÷ 5^{2}

Answer:

2560

Explanation:

8³ = 8 x 8 x 8 = 512

5^{3} = 5 x 5 x 5 = 125

5² = 5 x 5 = 25

8³ × 5^{3} ÷ 5² = 512 x 125 ÷ 25

= 2560

8³ × 5^{3} ÷ 5² = 2560

Question 17.

10^{3} ÷ 8^{2} × 4^{2}

Answer:

250

Explanation:

10³ = 10 x 10 x 10 = 1000

8² = 8 x 8 = 64

4² = 4 x 4 = 16

10³ ÷ 8² x 4² = 1000 ÷ 64 x 16

= 250

10³ ÷ 8² x 4² = 250.

Question 18.

7^{3} – 10^{2} ÷ 2^{2}

Answer:

318

Explanation:

7³ = 7 x 7 x 7 = 343

10² = 10 x 10 = 100

2² = 2 x 2 = 4

7³ – 10² ÷ 2² = 343 – (100 ÷ 4 )

= 343 – 25

= 318

7³ – 10² ÷ 2² = 318

Question 19.

3^{3} + 4^{3} × 6^{2}

Answer:

2331

Explanation:

3³ = 3 x 3 x 3= 27

4³ = 4 x 4 x 4 = 64

6² = 6 x 6 = 36

3^{3} + 4^{3} × 6² = 27 + (64 x 36 )

= 27 + 2304

= 2331

3^{3} + 4^{3} × 6² = 2331.

**Find the value of each of the following.**

Question 20.

17^{3}

Answer:

4913

Explanation:

I can relate this to finding the volume of a cube with edges of length 17 units.

17³ = 17 x 17 x 17

= 289 x 17

= 4913

The cube of 17 is 4913.

Question 21.

16^{3}

Answer:

4096

Explanation:

I can relate this to finding the volume of a cube with edges of length 16 units.

16³ = 16 x 16 x 16

= 256 x 16

= 4096

The cube of 16 is 4096.

Question 22.

18^{3}

Answer:

5832

Explanation:

I can relate this to finding the volume of a cube with edges of length 18 units.

18³ = 18 x 18 x 18

= 324 x 18

= 5832

The cube of 18 is 5832.

Question 23.

Answer:

12

Explanation:

Finding the cube root of a number is the inverse of finding the cube of a number

prime factorization of 1728 = 2 x 2x 2 x 2 x 2 x 2 x 3 x 3 x 3

= (2.2.3)³

=12³

So, cube root of 1728 is 12.

Question 24.

Answer:

20

Explanation:

Finding the cube root of a number is the inverse of finding the cube of a number

prime factorization of 8000 = 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 5

= (2.2.5)³

=20³

So, cube root of 8000 is 20.

Question 25.

Answer:

15

Explanation:

Finding the cube root of a number is the inverse of finding the cube of a number

prime factorization of 3375 = 3 x 3 x 3 x 5 x 5 x 5

= (3.5)³

=15³

So, cube root of 3375 is 15.

**Solve.**

Question 26.

Given that 11^{3} = 1,331, find the cube of 110.

Answer:

1,331,000

Explanation:

If 11^{3} = 1,331 then 110³ will be 1,331,000.

Question 27.

Given that 14^{3} = 2,744, find the cube root of 2,744,000.

Answer:

140³

Explanation:

If 14³ = 2,744 then cube root of 2,744,000 is 140³.

Question 28.

Given that = 16, evaluate 160^{3}.

Answer:

4,096,000

Explanation:

If cube root 4096 is 16 then 160³ will be 4,906,000.

Question 29.

Evaluate 13^{2} + 20^{3} – 18^{2}.

Answer:

7845

Explanation:

13² = 13 x 13 = 169

20³ = 20 x 20 x 20 = 8000

18² = 18 x 18 = 324

13^{2} + 20^{3} – 18² = 169 + 8000 – 324

= 7845

13^{2} + 20^{3} – 18² = 7845.

Question 30.

Evaluate the cube root of 12^{3} × 5^{3} + 8³.

Answer:

68

Explanation:

Cube root of 12^{3} × 5^{3} + 8³ will be the number itself

So, 12 x 5 + 8 = 68 is cube root of 12^{3} × 5^{3} + 8³.

Question 31.

Find three consecutive numbers whose cubes have a sum of 2,241.

Answer:

8, 9, 10

Explanation:

8³ = 512

9³ = 729

10³ = 1000

512 + 729 + 1000 = 2241

So, the three consecutive numbers whose cubes have a sum of 2241 are 8, 9 and 10.

Question 32.

A cubic crate with an edge length of 16 feet will be used to contain cubic wooden boxes. Each wooden box has an edge length of 4 feet. How many wooden boxes can the crate contain?

Answer:

4 wooden boxes

Explanation:

A cubic crate with an edge length of 16 feet will be used to contain cubic wooden boxes.

Each wooden box has an edge length of 4 feet

4 x 4 = 16

S0, the wooden crate can contain 4 wooden boxes.

**Brain @ Work**

Mr. Henderson wants to tile his patio that is rectangular in shape. His patio measures 108 inches by 144 inches. Find the fewest square tiles he can use without cutting any of them. (Hint: First find the largest size tile he can use.)

Answer:

21 tiles of 12 inches square tile

Explanation:

Mr. Henderson wants to tile his patio that is rectangular in shape.

His patio measures 108 inches by 144 inches

The largest size tile he can use is 12 inches

12 in x 12 tiles = 144 in

12 in x 9 tiles = 108 in

So, Mr. Henderson wants 21 tiles of 12 inches to tile his patio in rectangular shape without cutting any square tile.