Question

Simplify the expression. Then express your answer using rational indicators: \frac{\sqrt{xy}}{\sqrt[4]{16xy}}

Rational exponents and radicals
ANSWERED
asked 2021-07-30

Simplify the expression. Then express your answer using rational indicators. Let the letters stand for positive numbers.
\( \frac{\sqrt{xy}}{\sqrt[4]{16xy}}\)

Answers (1)

2021-07-31

Step 1
Formula:
\(\sqrt[n]{a^m}={a}^{{\frac{{m}}{{n}}}}\)
Law of exponent:
\(\displaystyle{\frac{{{a}^{{{m}}}}}{{{a}^{{{n}}}}}}={a}^{{{m}-{n}}}\)
Step 2
Use the property of n the root, in the expression \(\frac{\sqrt{xy}}{\sqrt[4]{16xy}}\)
\(\frac{\sqrt{xy}}{\sqrt[4]{16xy}}={\frac{{{\left({x}{y}\right)}^{{\frac{{1}}{{2}}}}}}{{{\left({16}\right)}^{{\frac{{1}}{{4}}}}{\left({x}{y}\right)}^{{\frac{{1}}{{4}}}}}}}\)
\(\displaystyle={\frac{{{\left({x}{y}\right)}^{{\frac{{1}}{{2}}}}}}{{{\left({2}^{{{4}}}\right)}^{{\frac{{1}}{{4}}}}{\left({x}{y}\right)}^{{\frac{{1}}{{4}}}}}}}\)
\(\displaystyle={\frac{{{\left({x}{y}\right)}^{{\frac{{1}}{{2}}}}}}{{{\left({2}\right)}^{{{4}\cdot{\frac{{{1}}}{{{4}}}}}}{\left({x}{y}\right)}^{{\frac{{1}}{{4}}}}}}}\)
\(\displaystyle={\frac{{{\left({x}{y}\right)}^{{\frac{{1}}{{2}}}}}}{{{2}{\left({x}{y}\right)}^{{\frac{{1}}{{4}}}}}}}\)
Now, use the law of exponents, in the above expression.
\(\displaystyle={\frac{{{\left({x}{y}\right)}^{{\frac{{1}}{{2}}}}}}{{{2}{\left({x}{y}\right)}^{{\frac{{1}}{{4}}}}}}}={\frac{{{\left({x}{y}\right)}^{{{\frac{{{1}}}{{{2}}}}-{\frac{{{1}}}{{{4}}}}}}}}{{{2}}}}\)
\(\displaystyle={\frac{{{\left({x}{y}\right)}^{{{\frac{{{2}-{1}}}{{{4}}}}}}}}{{{2}}}}\)
\(\displaystyle={\frac{{{\left({x}{y}\right)}^{{{\frac{{{1}}}{{{4}}}}}}}}{{{2}}}}\)
Thus, the solution of the expression \(\frac{\sqrt{xy}}{\sqrt[4]{16xy}}\) is \(\displaystyle={\frac{{{\left({x}{y}\right)}^{{\frac{{1}}{{4}}}}}}{{{2}}}}\)

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