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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