Find:
(i) $ 64^{\frac{1}{2}} $
(ii) $ 32^{\frac{1}{5}} $
(iii) $ 125^{\frac{1}{3}} $

To do:

We have to find the values of 
(i) \( 64^{\frac{1}{2}} \)
(ii) \( 32^{\frac{1}{5}} \)
(iii) \( 125^{\frac{1}{3}} \)
Solution: 

We know that,

$(a^m)^n=(a)^{mn}$

Therefore,

(i) $64^{\frac{1}{2}}=(8\times8)^{\frac{1}{2}}$

$=(8^2)^{\frac{1}{2}}$

$=(8)^{2\times\frac{1}{2}}$

$=8^1$

$=8$

Hence $64^{\frac{1}{2}}=8$

(ii) $32^{\frac{1}{5}}=(2\times2\times2\times2\times2)^{\frac{1}{5}}$

$=(2^5)^{\frac{1}{5}}$

$=(2)^{5\times\frac{1}{5}}$

$=2^1$

$=2$

Hence $32^{\frac{1}{5}}=2$

(iii) $125^{\frac{1}{3}}=(5\times5\times5)^{\frac{1}{3}}$

$=(5^3)^{\frac{1}{3}}$

$=(5)^{3\times\frac{1}{3}}$

$=5^1$

$=5$

Hence $125^{\frac{1}{3}}=5$

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Updated on: 10-Oct-2022

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