FIND CUBE ROOT OF A RATIONAL NUMBER

Evaluate:

Problem 1 :

∛(1/125)

Solution :

∛(1/125)

Distribute the cube root to numerator and denominator.

= ∛1 / ∛125

Decompose the number inside the radical sign into prime factors.

 = ∛(1 ∙ 1 ∙ 1) /∛(5 ∙ 5 ∙ 5)

Since the index is 3, we have to take one number out of radical sign for every three same numbers multiplied inside the radical sign.

= 1/5

Problem 2 :

∛(-8/27)

Solution :

∛(-8/27)

Distribute the cube root to numerator and denominator.

= ∛-8 / ∛27

Decompose the number inside the radical sign into prime factors.

 = - ∛(2 ∙ 2 ∙ 2) / ∛(3 ∙ 3 ∙ 3)

Since the index is 3, we have to take one number out of radical sign for every three same numbers multiplied inside the radical sign.

= -2/3

Problem 3 :

∛343/64

Solution :

∛(343/64)

Distribute the cube root to numerator and denominator.

= ∛(343 / 64)

Decompose the number inside the radical sign into prime factors.

 = ∛(7 ∙ 7 ∙ 7) / ∛(4 ∙ 4 ∙ 4)

Since the index is 3, we have to take one number out of radical sign for every three same numbers multiplied inside the radical sign.

= 7/4

Problem 4 :

-∛1  91/125

Solution :

= -∛1  91/125

= -∛(216/125)

Distribute the cube root to numerator and denominator.

= -∛(216 / 125)

Decompose the number inside the radical sign into prime factors.

 = -∛(6 ∙ 6 ∙ 6) / ∛(5 ∙ 5 ∙ 5)

Since the index is 3, we have to take one number out of radical sign for every three same numbers multiplied inside the radical sign.

= -6/5

Problem 5 :

∛(1/64)

Solution :

∛(1/64)

Distribute the cube root to numerator and denominator.

= ∛(1 / 64)

Decompose the number inside the radical sign into prime factors.

 = ∛(1 ∙ 1 ∙ 1) / ∛(4 ∙ 4 ∙ 4)

Since the index is 3, we have to take one number out of radical sign for every three same numbers multiplied inside the radical sign.

= 1/4

Problem 6 :

∛3  3/8

Solution :

= ∛3  3/8

Converting the mixed fraction into improper fraction, we get

= ∛27/8

Distribute the cube root to numerator and denominator.

= ∛27 / 8

Decompose the number inside the radical sign into prime factors.

 = ∛(3 ∙ 3 ∙ 3) / ∛(2 ∙ 2 ∙ 2)

Since the index is 3, we have to take one number out of radical sign for every three same numbers multiplied inside the radical sign.

= 3/2

Problem 7 :

∛-216

Solution :

∛-216

We can write 216 as 6 × 6 × 6,

∛-216 = ∛(- 6 × -6 × -6)

∛-216 = -6

Problem 8 :

∛27/125

Solution :

∛(27/125)

Distribute the cube root to numerator and denominator.

= ∛(27 / 125)

Decompose the number inside the radical sign into prime factors.

 = ∛(3 ∙ 3 ∙ 3) / ∛(5 ∙ 5 ∙ 5)

Since the index is 3, we have to take one number out of radical sign for every three same numbers multiplied inside the radical sign.

= 3/5

Problem 9 :

- ∛(343/1000)

Solution :

- ∛(343/1000)

Distribute the cube root to numerator and denominator.

= -∛(343 / 1000)

Decompose the number inside the radical sign into prime factors.

 = - ∛(7 ∙ 7 ∙ 7) / ∛(10 ∙ 10 ∙ 10)

Since the index is 3, we have to take one number out of radical sign for every three same numbers multiplied inside the radical sign.

= -7/10

Problem 10 :

∛729/64

Solution :

∛(729/64)

Distribute the cube root to numerator and denominator.

∛(729 / 64)

Decompose the number inside the radical sign into prime factors.

 = ∛(9 ∙ 9 ∙ 9) / ∛(4 ∙ 4 ∙ 4)

Since the index is 3, we have to take one number out of radical sign for every three same numbers multiplied inside the radical sign.

= 9/4

Problem 11 :

∛(0.027/0.008) ÷ √(0.09/0.04) - 1

Solution :

= ∛(0.027/0.008) ÷ √(0.09/0.04) - 1

∛(0.027/0.008) = ∛0.027/∛0.008

0.027 = 27/1000

0.008 = 8/1000

Dividing these two fractions, we get

= 27/1000 / (8/1000)

= (27 / 1000) x (1000 / 8)

= 27/8

√(0.09/0.04) = √0.09/√0.04

0.09 = 9/100

0.04 = 4/100

Dividing these two fractions, we get

= 9/100 / (4/100)

= 9/100 x 100/4

= 9/4

Applying these values, we get

= 27/8 ÷ (9/4) - 1

= 27/8 x (4/9) - 1

= 3/2 - 1

= (3 - 2)/2

= 1/2

Problem 12 :

Find the volume of a cube whose surface area is 150 m²

Solution :

Surface area of cube = 150 m²

Let a be the side length of cube.

6a² = 150

a² = 150/6

a² = 25

a = 5

Volume of cube = a³

= 5³

= 125 m³

Problem 13 :

Evaluate the cube root of 686 / 1024

Solution :

= ∛(686 / 1024)

= ∛686 /  ∛1024

= ∛(2 x 7 x 7 x 7) /  ∛(2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2)

= 7∛2 / (2 x 2 x 2) ∛2

= 7/8

Problem 14 :

What is the smallest number by which 4608 may be multiplied so that the product is perfect cube?

Solution :

Writing 4608 as product of prime numbers,  we get

=  ∛(2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 2 x 2 x 3)

After factoring one 2 for every three same values, we get

= 2 x 2 x 2  ∛(3 x 3)

= 8∛(3 x 3)

Since we need one more three, to make it as perfect cube. So, the required number is 3.

Problem 15 :

Find the surface area of a cube whose volume is 343 m³

Solution :

Let a be the side length of cube.

volume = 343 m³

a³ = 343

a = ∛343

a = ∛(7 x 7 x 7)

= 7

So, side length of the cube is 7 m.

Surface area of cube = 6 a²

= 6(7²)

= 6(49)

= 294 m²