Solve the following:$\sqrt{3\times 5^{-3} \ } \div \sqrt[3] 3^{-1} \ \sqrt{5} \ \times \acute{}\sqrt[6]{3\times 5^{5}}$

We know that,

$\begin{array}{l}$
a^{-m} =\frac{1}{a^{m}}\\
a^{m} \times a^{n} =a^{m+n}\\
\frac{a^{m}}{a^{n}} =a^{m-n}
\end{array}

$\sqrt[m]{a} =( a)^{\frac{1}{m}}$

$\begin{array}{l}$
\sqrt{3\times 5^{-3}} \div \sqrt[3]{3^{-1}}\sqrt{5} \times \sqrt[6]{3\times 5^{6}} =\left( 3\times 5^{-3}\right)^{\frac{1}{2}} \div \left( 3^{-1}\right)^{\frac{1}{3}}( 5)^{\frac{1}{2}} \times \left( 3\times 5^{6}\right)^{\frac{1}{6}}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =( 3)^{\frac{1}{2}} \times \left( 5^{-3}\right)^{\frac{1}{2}} \div \left( 3^{-1}\right)^{\frac{1}{3}}( 5)^{\frac{1}{2}} \times ( 3)^{\frac{1}{6}} \times \left( 5^{6}\right)^{\frac{1}{6}}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =( 3)^{\frac{1}{2}} \times ( 5)^{-3\times \frac{1}{2}} \div ( 3)^{-1\times \frac{1}{3}}( 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}}( 5)^{\frac{1}{2}} \times ( 3)^{\frac{1}{6}} \times ( 5)^{1}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{( 3)^{\frac{1}{2}} \times ( 5)^{\frac{-3}{2}}}{( 3)^{\frac{-1}{3}}( 5)^{\frac{1}{2}}} \times ( 3)^{\frac{1}{6}} \times 5\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =( 3)^{\frac{1}{2} -\left( -\frac{1}{3}\right) +\frac{1}{6}} \times ( 5)^{\frac{-3}{2} -\frac{1}{2} +1}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =( 3)^{\frac{1\times 3+1\times 2+1}{6}} \times ( 5)^{\frac{-3-1+1\times 2}{2}}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =( 3)^{\frac{3+2+1}{6}} \times ( 5)^{\frac{-4+2}{2}}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =( 3)^{\frac{6}{6}} \times ( 5)^{\frac{-2}{2}}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =3\times 5^{-1}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{3}{5}\\
\end{array}

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Updated on: 10-Oct-2022

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