# Cube Root of Rational Numbers

Definition of cube root of rational numbers : If x and a are two rational numbers such that x 3 = a then we say that x is the cube root of a and we write
∛a = x

We know that,
(3/4)
3 = 27/64 ⇒ ∛(27/64) = 3/4

(-4/11)
3 = -64/1331 ⇒ ∛(-64/1331) = -4/11

So in order to find the cube-root of rational-numbers we use the following :
For any rational number a/b, we have

∛(a/b) = (∛a)/(∛b)

Examples :

1) ∛(27/8)

Solution :

∛(27/8) = (∛27)/(∛8)

= (∛3 x 3 x 3)/(∛2 x 2 x 2 )

∛(27/8) = 3/2
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2) ∛(-64/125)

Solution :

∛(-64/125) = (∛-64)/(∛125)

= (∛-4 x -4 x -4)/(∛5 x 5 x 5 )

∛(-64/125) = -4/5
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3) ∛(1.331)

Solution :

∛(1.331) = ∛(1331/1000)

∛(1331/1000) = (∛1331)/(∛1000)

= (∛11 x 11 x 11)/(∛10 x 10 x 10 )

∛(1.331) = 11/10

∛(1.331) = 1.1
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4) ∛(0.008)

Solution :

∛(0.008) = ∛(8/1000)

∛(8/1000) = (∛8)/(∛1000)

= (∛2 x 2 x 2)/(∛10 x 10 x 10 )

∛(0.008) = 2/10

∛(0.008)= 0.2
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5)∛(0.125)

Solution :

∛(0.125) = ∛(125/1000)

∛(125/1000) = (∛125)/(∛1000)

= (∛5 x 5 x 5)/(∛10 x 10 x 10 )

∛(0.125) = 5/10

∛(0.125)= 0.5
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6)∛(216/343)

Solution :

∛(216/343) = (∛216)/(∛343)

= (∛6 x 6 x 6)/(∛7 x 7 x 7 )

∛(216/343) = 6/7
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7)∛(1000/8)

Solution :

∛(1000/8) = (∛1000)/(∛8)

= (∛10 x 10 x 10)/(∛2 x 2 x 2 )

∛(1000/8) = 10/8

∛(1000/8)= 5/4

Cube and Cube Roots

Cube of Numbers
Perfect Cube
Properties of Cube
Cube by Column method
Cube of Negative numbers
Cube of Rational numbers
Cube Root
Finding cube root by Prime Factorization
Cube root of Rational numbers
Estimating cube root

From Cube root of rational-numbers to Exponents