Question

$\large\underline{ \underline{ \red{ \tt{Question :}}}}$

Given that :-

$\sf \: ^{3}\sqrt{3^{x}} = 5^{ \frac{1}{4} } \: \: and \: \: ^{4}\sqrt{5^{y}} = \sqrt{3}$

Then, find the value of 2xy.

$\underline{ \underline{ \tt{Note :}}}$

• Need proper explanation.
• No spammed answer.

All the best! ​

Answer 1

❥Given :

• $\pmb{\blue{\sqrt[3]{ {3}^{x} } }} = \pmb{\green{ 5 ^\frac{1}{4} }}$
• $\pmb{\orange{ \sqrt[4]{ {5}^{y} } }} = \pmb{\pink{ \sqrt{3} }}$

❥To find :

• Finding value of 2xy

❥Solution :

$\pmb{1){\blue{ \sqrt[3]{ {3}^{x} } }} = {\green{5 ^ \frac{1}{4} }}}$

$\Rightarrow \pmb{\blue{( {{3}^{x}) \frac{1}{3} \: = \: } }} \pmb{\green{5 ^\frac{1}{4} }}$

$\Rightarrow \pmb{\blue{3 ^\frac{x}{3} }} = \pmb{\green{5 ^ \frac{1}{4} }}$

$\Rightarrow\pmb{\blue{(3 ^ \frac{x}{3} )^{4} }} = \pmb{\green{(5 ^\frac{1}{\cancel{{4}}} \pink{1} )^{\cancel{{4}}} \pink{1} }}$

$\boxed{\Rightarrow \pmb{\blue{3 ^ \frac{4x}{3} = \pmb{\green{5}}}}}$

$\pmb{2){\orange{ \sqrt[4]{ {5}^{y} } }} = \pmb{\pink{ \sqrt{3} }}}$

$\Rightarrow \pmb{{\orange{( {5}^{y} ) \frac{1}{4} }} = {\pink{3 ^\frac{1}{2} }}}$

$\Rightarrow \pmb{\orange{(5 ^ \frac{y}{4}) } = {\pink{3 ^ \frac{1}{2} }}}$

$\Rightarrow \pmb{{\orange{(3 ^ \frac{4x}{3}) ^ \frac{y}{4} }} ={\pink{3 ^ \frac{1}{2} }}}$

$\Rightarrow\pmb{{\orange{3 ^ \frac{4xy}{12} }} = {\pink{3 ^ \frac{1}{2} }}}$

Now, as the bases are same so powers are also same so

$\Rightarrow \pmb{{\orange{ \frac {\cancel{{4xy}} \: \green{xy}}{\cancel{{12}} \: \green{3}}}} \nearrow {\pink{ \frac{1}{2} }} = {\orange{\frac{xy}{3}} \searrow} {\pink{ \frac{1}{2} }}}$

$\boxed {\therefore{ \pmb{\purple{2xy = { \gray{3}}}}}}$

Answer 2

Solution!!

Let's solve for x!

$\sf \to \sqrt[3]{3^{x}}=5^{\frac{1}{4}}$

We know that $\sf a^{\frac{m}{n}}=\sqrt[n]{a^{m}}$

$\sf \to \sqrt[3]{3^{x}}=\sqrt[4]{5}$

Raising to the power of 3

$\sf \to 3^{x}=\sqrt[4]{125}$

Taking the logarithm

$\sf \to \log_{3}(3^{x})=\log_{3}(\sqrt[4]{125})$

We know that $\sf \log_{a}(a^{x})=x$

$\sf \to x=\log_{3}(\sqrt[4]{125})$

Writing in the exponential form

$\sf \to x=\log_{3}(5^{\frac{3}{4}})$

We know that $\sf \log_{a}(b^{c})=c\times \log_{a}(b)$

$\sf \to x=\dfrac{3}{4}\times \log_{3}(5)$

Let's solve for y!

$\sf \to \sqrt[4]{5^{y}}=\sqrt{3}$

Raising to the power of 4

$\sf \to 5^{y}=9$

Taking the logarithm

$\sf \to \log_{5}(5^{y})=\log_{5}(9)$

We know that $\sf \log_{a}(a^{x})=x$

$\sf \to y=\log_{5}(9)$

Writing in the exponential form

$\sf \to y=\log_{5}(3^{2})$

We know that $\sf \log_{a}(b^{c})=c\times \log_{a}(b)$

$\sf \to y=2\log_{5}(3)$

Now, we'll find the value of 2xy!

$\sf \to 2xy=2\times \dfrac{3}{4}\times \log_{3}(5)\times 2\log_{5}(3)$

$\sf \to 2xy=\dfrac{3}{2}\times \log_{3}(5)\times 2\log_{5}(3)$

$\sf \to 2xy=3\log_{3}(5)\log_{5}(3)$

We know that $\sf \log_{b}(a)\times \log_{a}(b)=1$

$\sf \to 2xy=3\times 1$

$\sf \to 2xy=3$