Eight Standard >> Cube and cube roots | Part -2

Find the cube roots of a number

Cube roots are the reverse process of the cube, like 
  2 cubed=\(2^{3}\)=8 and cube root of 8=2
  3 cubed=\(3^{3}\)=27 and cube root of 27=3

Cube root of a number 'a' is denoted by 
  \(\sqrt[3]{a}\) or \((a)^{\frac{1}{3}}\)
So, \(\sqrt[3]{8}\)= \(\sqrt[3]{(2 \times 2 \times 2}\)=2

The cube root of a number 'a' is one of three equal factors.
   \(\sqrt[3]{8}\)= \(\sqrt[3]{(2 \times 2 \times 2}\)
             =2(Taking one factor 2)
Number    Cube of the number    Cube roots of the number 
1    \(1^{3}\)    \(\sqrt[3]{1}\)= 1
2    \(2^{3}\)    \(\sqrt[3]{8}\)=2
3    \(3^{3}\)    \(\sqrt[3]{27}\)=3
\(\frac{1}{2}\)    \((\frac{1}{2})^{3}\)    \(\sqrt[3]{\frac{1}{2}}\) =\(\frac{1}{2}\)
-1    \((-1)^{3}\)=-1    \(\sqrt[3]{(-1)}\)=-1
-2    \((-2)^{3}\)=-8    \(\sqrt[3]{(-8)}\)=-2

In the case of square root, we can't take '-'. Because it's not a real number but in the case of cube root we can take '-'.

How to find the cube roots of a number.

Step 1: Do the prime factorization of the given number.
Step 2: Write the prime factors in index notation.
Step 3: Divide the index by 3 and the resulting number is the cube root of the number.

Example: Find the cube root of 216

Step 1:
  216=\(2 \times 2 \times 2 \times 3 \times 3 \times 3\)
Step 2:
  216=\(2^{3}\times 3^{3}\)
Step 3:
  \((216)^{\frac{1}{3}}\)=\(\Big(2^{3}\Big)^{\frac{1}{3}} \times \Big(3^{3}\Big)^{\frac{1}{3}}\)
      =2\(\times\)3=6