Question

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

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 $$\displaystyle={\frac{{{\left({x}{y}\right)}^{{\frac{{1}}{{4}}}}}}{{{2}}}}$$