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Question of The Day24-10-2019

Arrange the following in ascending order

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

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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