Question

arrange
$\sqrt[4]{9} \: \sqrt[3]{5} \: \sqrt[6]{26}$
in descending order​

Answer 1

$\huge\red{\mathbb {\underline {Answer}}}$

$\sqrt[4]{9} = {(9)}^{ \frac{1}{4}}\\ \sqrt[3]{5} = {(5)}^{ \frac{1}{3} } \\ \sqrt[6]{26} = {(26)}^{ \frac{1}{6} }$

$\small\sf\blue{We \: will \: take \: the \: LCM \: of \: 4, 3 \: and \: 6 }$

$\small\sf\blue{LCM \: of \: 4, 3 \: and \: 6=12}$

${(9)}^{ \frac{1}{4}} = {9}^{( \frac{1 \times 3}{4 \times 3} = \frac{3}{12} )} = {(9)}^{ \frac{3}{12} } \\ {(5)}^{\frac{1}{3}} = {5}^{( \frac{1 \times 4}{3 \times 4} = \frac{4}{12} )} = {(5)}^{ \frac{4}{12} } \\ {(26)}^{ \frac{1}{6} } = {26}^{( \frac{1 \times 2}{6 \times 2} = \frac{2}{12} )} = {(26)}^{ \frac{2}{12} }$

$\small\sf\blue{Using \: the \: formula,}$

$\small\sf\blue{ \sqrt[m]{a} = {a}^{ \frac{1}{m} }}$

$\small\sf\blue{ we \: get ,}$

${(9)}^{ \frac{3}{12} } = {(9)}^{3 \times \frac{1}{12} } = \sqrt[12]{ {(9)}^{3} } = \sqrt[12]{729} \\{(5)}^{ \frac{4}{12} } = {(5)}^{4 \times \frac{1}{12} } = \sqrt[12]{ {(5)}^{4} } = \sqrt[12]{625} \\{(26)}^{ \frac{2}{12} } = {(26)}^{2 \times \frac{1}{12} } = \sqrt[12]{ {(26)}^{2} } = \sqrt[12]{676}$

$\small\sf\blue{Comparing \: the \: numbers,we \: get,}$

$\small\sf\orange{ \sqrt[12]{729} > \sqrt[12]{676} > \sqrt[12]{625} }$

$\small\sf\green{HENCE, \: \sqrt[4]{9} > \sqrt[6]{2} > \sqrt[3]{5} }$

Answer 2

$\huge{ \underline{ \bold{ᴀɴsᴡᴇʀ....{ \heartsuit}}}}$

$$\begin{lgathered}\sqrt[4]{9} = {(9)}^{ \frac{1}{4}}\\ \sqrt[3]{5} = {(5)}^{ \frac{1}{3} } \\ \sqrt[6]{26} = {(26)}^{ \frac{1}{6} }\end{lgathered}$$

$$\small\sf\blue{We \: will \: take \: the \: LCM \: of \: 4, 3 \: and \: 6 }$$

$$\small\sf\blue{LCM \: of \: 4, 3 \: and \: 6=12}$$

$$\begin{lgathered}{(9)}^{ \frac{1}{4}} = {9}^{( \frac{1 \times 3}{4 \times 3} = \frac{3}{12} )} = {(9)}^{ \frac{3}{12} } \\ {(5)}^{\frac{1}{3}} = {5}^{( \frac{1 \times 4}{3 \times 4} = \frac{4}{12} )} = {(5)}^{ \frac{4}{12} } \\ {(26)}^{ \frac{1}{6} } = {26}^{( \frac{1 \times 2}{6 \times 2} = \frac{2}{12} )} = {(26)}^{ \frac{2}{12} }\end{lgathered}$$

$$\small\sf\blue{Using \: the \: formula,}$$

$$\small\sf\blue{ \sqrt[m]{a} = {a}^{ \frac{1}{m} }}$$

$$\small\sf\blue{ we \: get ,}$$

$$\begin{lgathered}{(9)}^{ \frac{3}{12} } = {(9)}^{3 \times \frac{1}{12} } = \sqrt[12]{ {(9)}^{3} } = \sqrt[12]{729} \\{(5)}^{ \frac{4}{12} } = {(5)}^{4 \times \frac{1}{12} } = \sqrt[12]{ {(5)}^{4} } = \sqrt[12]{625} \\{(26)}^{ \frac{2}{12} } = {(26)}^{2 \times \frac{1}{12} } = \sqrt[12]{ {(26)}^{2} } = \sqrt[12]{676}\end{lgathered}$$

$$\small\sf\blue{Comparing \: the \: numbers,we \: get,}$$

$$\small\sf\orange{ \sqrt[12]{729} > \sqrt[12]{676} > \sqrt[12]{625} }$$

$$\small\sf\green{HENCE, \: \sqrt[4]{9} > \sqrt[6]{2} > \sqrt[3]{5} }$$