Simplify the expression. Express your answer using rational indicators if the letters represent positive numbers: \frac{\sqrt{a^{3}b}}{\sqrt[4]{a^{3}b^{2}}}

Simplify the expression. Express your answer using rational indicators if the letters represent positive numbers.
$$\frac{\sqrt{a^3b}}{\sqrt[4]{a^3b^2}}$$

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Sally Cresswell

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 $$\displaystyle{a}^{{\frac{{3}}{{4}}}}$$

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