nth Root of a

We will discuss here about the meaning of \(\sqrt[n]{a}\).

The expression \(\sqrt[n]{a}\) means ‘nth rrot of a’. So, (\(\sqrt[n]{a}\))^n = a.

Also, (a1/a)n = a n × 1/n = a1 = a.

So, \(\sqrt[n]{a}\) = a1/n.


1. \(\sqrt[3]{8}\) = 81/3

          = (23)1/3

          = 23 × 1/3

          = 21

          = 2.

2. \(\sqrt[4]{9}\) = 91/4

          = (32)¼   

          = 32 × ¼

          = 31/2

          = √3.

Note: 31/2 = \(\sqrt[2]{3}\). But \(\sqrt[2]{3}\) is also written as √3.

nth Root of a

Solved examples on nth Root of a:

Express each of the following in the simplest form without radicals:

(i) \(\sqrt[4]{5^{2}}\)

(ii) \(\sqrt[n]{x^{m}}\)

(iii) \(\sqrt[3]{64^{-4}}\)


(i) \(\sqrt[4]{5^{2}}\) = (52)1/4

             = 52 × 1/4

(ii) \(\sqrt[n]{x^{m}}\) = (xm)1/n

              = xm × 1/n

              = xm/n.

(iii) \(\sqrt[3]{64^{-4}}\) = (64-4)1/3

                 = 64-4 × 1/3

                 = 64-4/3

                 = (43)-4/3

                 = 43(-4/3)

                 = 4-4

                 = \(\frac{1}{4 × 4 × 4 × 4}\)

                 = \(\frac{1}{256}\).

9th Grade Math

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