# 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.

Examples:

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.

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

Solution:

(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

= $$\frac{1}{4 × 4 × 4 × 4}$$

= $$\frac{1}{256}$$.