QOTD | Quantitative Aptitude | Surds and Indices

Question of The Day24-10-2019

Arrange the following in ascending order

$ \sqrt[5]{(2)^{390}} < \sqrt{(9)^{65}} < \sqrt[3]{(7)^{117}}$

Answer

Correct Answer : c ) $ \sqrt[5]{(2)^{390}} < \sqrt{(9)^{65}} < \sqrt[3]{(7)^{117}}$

Explanation :

According to the law of surds

$\sqrt[m]{\sqrt[n]{\sqrt[o]{(((x)^{p})^{q})^{r}}}}$can be written as $x^{(\frac{P*q*r}{m*n*o})}$

Thus,

$\sqrt[5]{(2)^{390}}=2^{\frac{390}{5}}=2^{78}..............1$

Similarly,

$\sqrt{(9)^{65}}=9^{\frac{65}{2}}=3^{65}..............2$

Similarly,

$\sqrt[3]{(7)^{117}}=7^{\frac{117}{3}}=7^{39}..............3$

Now, equating the powers in equation 1, 2, and 3, we get

${(2)^{78}} =>{(64)^{13}} $

${(3)^{65}} => {(243)^{13}}$

${(7)^{39}} => {(343)^{13}}$

Therefore, the correct order will be

${(64)^{13}} < {(243)^{13}} < {(343)^{13}}$

$\sqrt[5]{(2)^{390}} < \sqrt{(9)^{65}} < \sqrt[3]{(7)^{117}}$

Hence, (C) is the correct answer.

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