Prove that:$ \sqrt{3 \times 5^{-3}} \p \sqrt[3]{3^{-1}} \sqrt{5} \times \sqrt[6]{3 \times 5^{6}}=\frac{3}{5} $

Given:

\( \sqrt{3 \times 5^{-3}} \div \sqrt[3]{3^{-1}} \sqrt{5} \times \sqrt[6]{3 \times 5^{6}}=\frac{3}{5} \)

To do:

We have to prove that \( \sqrt{3 \times 5^{-3}} \div \sqrt[3]{3^{-1}} \sqrt{5} \times \sqrt[6]{3 \times 5^{6}}=\frac{3}{5} \).

Solution:

We know that,

$(a^{m})^{n}=a^{m n}$

$a^{m} \times a^{n}=a^{m+n}$

$a^{m} \div a^{n}=a^{m-n}$

$a^{0}=1$

Therefore,

LHS $=\sqrt{3 \times 5^{-3}} \div \sqrt[3]{3^{-1}} \sqrt{5} \times \sqrt[6]{3 \times 5^{6}}$

$=(3 \times 5^{-3})^{\frac{1}{2}} \div(3^{-1})^{\frac{1}{3}}(5)^{\frac{1}{2}} \times(3 \times 5^{6})^{\frac{1}{6}}$

$=(3^{\frac{1}{2}} \times 5^{\frac{-3}{2}}) \div(3^{\frac{-1}{3}} \times 5^{\frac{1}{2}}) \times(3^{\frac{1}{6}} \times 5^{6 \times \frac{1}{6}})$

$=(3^{\frac{1}{2}} \times 5^{\frac{-3}{2}}) \div(3^{\frac{-1}{3}} \times 5^{\frac{1}{2}}) \times 3^{\frac{1}{6}} \times 5^{1}$

$=3^{\frac{1}{2}-(\frac{-1}{3})+\frac{1}{6}} \times 5^{\frac{-3}{2}-\frac{1}{2}+1}$

$=3^{\frac{1}{2}+\frac{1}{3}+\frac{1}{6}} \times 5^{\frac{-3-1+2}{2}}$

$=\frac{3+2+1}{6} \times 5^{\frac{-4+2}{2}}$

$=3^{\frac{6}{6}} \times 5^{\frac{-2}{2}}$

$=3^{1} \times 5^{-1}$

$=\frac{3}{5}$

$=$ RHS

Hence proved.