Step 1

Use the property of n the root, in the expression

\(\frac{\sqrt{a^3b}}{\sqrt[4]{a^3b^2}}={\frac{{{a}^{{\frac{{3}}{{2}}}}{b}^{{\frac{{1}}{{2}}}}}}{{{a}^{{\frac{{3}}{{4}}}}{b}^{{\frac{{2}}{{4}}}}}}}\)

\(\displaystyle={\frac{{{a}^{{\frac{{3}}{{2}}}}{b}^{{\frac{{1}}{{2}}}}}}{{{a}^{{\frac{{3}}{{4}}}}{b}^{{\frac{{1}}{{2}}}}}}}\)

Now, use the law of exponents, in the above expression.

\(\displaystyle{\frac{{{a}^{{\frac{{3}}{{2}}}}{b}^{{\frac{{1}}{{2}}}}}}{{{a}^{{\frac{{3}}{{4}}}}{b}^{{\frac{{1}}{{2}}}}}}}={a}^{{{\left({\frac{{{3}}}{{{2}}}}-{\frac{{{3}}}{{{4}}}}\right)}}}{b}^{{{\left({\frac{{{1}}}{{{2}}}}-{\frac{{{1}}}{{{2}}}}\right)}}}\)

\(\displaystyle={a}^{{{\left({\frac{{{6}-{3}}}{{{4}}}}\right)}}}{b}^{{{\left({\frac{{{1}-{1}}}{{{2}}}}\right)}}}\)

\(\displaystyle={a}^{{{\frac{{{3}}}{{{4}}}}}}{b}^{{{\frac{{{0}}}{{{2}}}}}}\)

\(\displaystyle={a}^{{{\frac{{{3}}}{{{4}}}}}}\)

Thus, the solution of the expression \(\frac{\sqrt{a^3b}}{\sqrt[4]{a^3b^2}}\) is \(\displaystyle{a}^{{\frac{{3}}{{4}}}}\)