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}}}

illusiia 2021-07-30 Answered

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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Expert Answer

Sally Cresswell
Answered 2021-07-31 Author has 14427 answers

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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Answered 2021-10-29 Author has 11829 answers

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