In the following exercises, write with a rational exponent.

Part (a): $\sqrt{m^5}$

Part (b):${\left(\sqrt[3]{3y}\right)}^7$

Part (c):$\sqrt[5]{{\left(\frac{4x}{5y}\right)}^3}$

Consider the given expression,

$\sqrt{m^5}$

We want to use localid="1644816988670" $a^{\frac{m}{n}}=\sqrt[n]{a^m}$to write each radical in the form $a^{\frac{m}{n}}$.

The numerator of the exponent is the exponent, $5$.

The denominator of the exponent is the index of the radical, $2$.

Therefore, the answer is$m^{\frac{5}{2}}$.

Consider the given expression,

${\left(\sqrt[3]{3y}\right)}^7$

We want to use $a^{\frac{m}{n}}=\sqrt[n]{a^m}$to write each radical in the form $a^{\frac{m}{n}}$.

The numerator of the exponent is the exponent, $7$.

The denominator of the exponent is the index of the radical, $3$.

Therefore, the answer is role="math" localid="1644817811370" ${\left(3y\right)}^{\frac{7}{3}}$.

Consider the given expression,

$\sqrt[5]{{\left(\frac{4x}{5y}\right)}^3}$

We want to use $a^{\frac{m}{n}}=\sqrt[n]{a^m}$to write each radical in the form $a^{\frac{m}{n}}$.

The numerator of the exponent is the exponent, $3$.

The denominator of the exponent is the index of the radical, $5$.

Therefore, the answer is ${\left(\frac{4x}{5y}\right)}^{\frac{3}{5}}$.