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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
= \(\frac{1}{4 × 4 × 4 × 4}\)
= \(\frac{1}{256}\).
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