Finding The Cube Root Of Any Given Number

Volume of a cube is the product of its edges. In a cube, all the sides are of equal length. Hence, if one of the edges is known then the volume can be easily calculated. Volume of cube = \(side~\times~side~\times~side\) = \(side^3\). Now, what if volume of the cube is given and we have to calculate the length of its edge. This can be easily done by finding the cube root of the volume. The value obtained after calculating the cube root will give us its edge length. For example, the volume of a cube is 125. Its length will be:

\(125\) = \(a^3\),
\(\Rightarrow 5~\times~5~\times~5\) = \(a^3\)
\(\Rightarrow~a\) = \(5\)

Therefore the side of the cube is 5 units.

Methods for calculating the cube root of a number

Addition is the inverse of subtraction and multiplication is the inverse function of division. Similarly cube root is the inverse of a cube number. For example,

\(1^3\) = \(1\), the cube root of 1 is 1
\(4^3\) = \(64\), cube root of 64 is 4 and so on.

Please note that we will only consider the positive values cube roots of the natural numbers. We can write \(\sqrt[3]{125}\) = \(5\) and \(\sqrt[3]{64}\) = \(4\). The symbol \(\sqrt[3]{}\) denotes the cube root.

Prime Factorization

We can find the cube root of a number by the method of prime factorization . Consider the following example for a clear understanding:

\(2744\) = \(2~\times~2~\times~2~\times~7~\times~7~\times~7\) = \((2~×~7)^3\)

Therefore the cube root of \(2744\) = \(∛2744\) = \(2~×~7\) = \(14\)

Illustration 1: Solve: \(\sqrt[3]{24389}\).
Solution: Prime factors = \(29~×~29~×~29\) = \(293\)
Therefore \(\sqrt[3]{24389}\) = \(29\).

Illustration 2: Find \(\sqrt[3]{46656}\) by the method of prime factorization.

Solution: Let us first find the prime factors:

Cube Root

\(46656\) = \(2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3\) = \(23 × 23 × 33 × 33\) = \((2 × 2 × 3 × 3)^3\)

Therefore, \(\sqrt[3]{46656}\) = \(36\)<

Learn more about the concept in depth through NCERT Solutions which is the latest edition from team Byju’s. Get detailed solutions to the questions of NCERT Books for chapter Cube and Cube Roots along with detailed explanation.


Practise This Question

The value of 30.1×0.1×0.1×13×13×13