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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.
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