CUBE ROOT OF THE GIVEN NUMBER

Evaluate each of the following:

Problem 1 :

∛-27

Solution :

= ∛-27

We can write 27 as 3 × 3 × 3,

∛-27 = ∛ - (3 × 3 × 3)

∛-27 = -3

Problem 2 :

∛1/8

Solution :

= ∛1/8

= ∛1/ ∛8

= 1/2

Problem 3 :

∛27

Solution :

= ∛27

We can write 27 as 3 × 3 × 3,

∛27 = ∛ (3 × 3 × 3)

∛27 = 3

Problem 4 :

- ∛1/64

Solution :

= -∛1/64

= -∛1/ ∛64

= -1/4

Problem 5 :

∛-8/27

Solution :

= ∛-8/27

= ∛-8/ ∛27

= -2/3

Problem 6 :

∛-1000

Solution :

= ∛-1000

We can write 1000 as 10 × 10 × 10,

∛-1000 = -∛ (10 × 10 × 10)

∛-1000 = -10

Problem 7 :

∛-27/64

Solution :

= ∛-27/64

= ∛-27/ ∛64

= -3/4

Problem 8:

-∛-64

Solution :

= -∛-64

We can write 64 as 4 × 4 × 4,

-∛-64 = -∛ -(4 × 4 × 4)

-∛-64 = 4

Problem 9 :

∛-1/125

Solution :

= ∛-1/125

= ∛-1/ ∛125

= -1/5

Problem 10 :

The volume of a cube-shaped compost bin is 27 cubic feet. What is the edge length of the compost bin?

Solution :

Volume of cube = 27 cubic feet

Let x be the side length of cube.

x³ = 27

x = ∛27

x = ∛(3 ⋅ 3 ⋅ 3)

x = 3

So, side length of compost bin is 3 ft.

Problem 11 :

The volume of a cube of ice for an ice sculpture is 64,000 cubic inches.

a. What is the edge length of the cube of ice?

b. What is the surface area of the cube of ice?

Solution :

Volume of cube = 64,000 cubic inches

Let x be the side length of cube, then x³ = 64000

x = ∛64000

x = ∛(40 ⋅ 40 ⋅ 40)

x = 40 inches

a) Side length of the cube is 40 inches

b) Surface area of cube = 6a²

= 6(40)²

= 6(1600)

= 9600 square inches.

Problem 12 :

Solve the equation.

a) (3x + 4)³ = 2197

b) (8x³ − 9)³ = 5832

c) ((5x − 16)³ − 4)³ = 216,000

Solution :

a) (3x + 4)³ = 2197

3x + 4 = ∛2197

3x + 4 = ∛(13 ⋅ 13 ⋅13)

3x + 4 = 13

3x = 13 - 4

3x = 9

x = 9/3

x = 3

b) (8x³ − 9)³ = 5832

(8x³ − 9) = ∛5832

(8x³ − 9) = ∛(18 ⋅ 18 ⋅ 18)

(8x³ − 9) = 18

8x³ = 18 + 9

8x³ = 27

x³ = 27/8

x = ∛(27/8)

x = 3/2

c) ((5x − 16)³ − 4)³ = 216,000

(5x − 16)³ − 4 = ∛216000

(5x − 16)³ − 4 = ∛(60 ⋅ 60 ⋅ 60)

(5x − 16)³ − 4 = 60

(5x − 16)³ = 60 + 4

(5x − 16)³ = 64

5x - 16 = ∛64

5x - 16 = 4

5x = 4 + 16

5x = 20

x = 20/5

x = 4

Problem 13 :

You bake a dessert in the baking pan shown. You cut the dessert into cube-shaped pieces of equal size. Each piece has a volume of 8 cubic inches. How many pieces do you get from one pan? Justify your answer.

Solution :

Volume of each piece = 8 cubic inches

Side length of each piece = ∛8

= 2 inches

Area of square base = 64 square inches

Number pieces are placed = 64/2

= 32 pieces.

Problem 14 :

The formula for volume of a pyramid is V = (1/3)ℓwh. The pyramid has a volume of 972 cubic inches. What are the dimensions of the pyramid?

Solution :

Volume of cube = 972 cubic inches

V = (1/3)ℓwh

Length = x inches, width = x inches and height = x/2

972 = (1/3) x ⋅ x ⋅ (x/2)

972 = (1/6) x³

972(6) = x³

x = ∛972(6)

x = 18

Height = 18/2 ==> 9 inches

So, length = 18 inches, width = 18 inches and height = 9 inches

Problem 15 :

There are three numbers that are their own cube roots. What are the numbers?

Solution :

• Cube root of -1 = ∛(-1) ==> -1
• Cube root of 1 = ∛1 ==> 1
• Cube root of 0 = ∛0 ==> 0

So, -1, 0 and 1 are the numbers which has their own cube roots.