FIND CUBE ROOTS 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