Q 141E - Intermediate Algebra

Question

In the following exercises, write with a rational exponent.

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

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

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

Step-by-Step Solution

Answer

Part (a): On writing with a rational exponent is $m^{\frac{5}{2}}$.

Part (b): On writing with a rational exponent is ${\left(3y\right)}^{\frac{7}{3}}$.

Part (c): On writing with a rational exponent is ${\left(\frac{4x}{5y}\right)}^{\frac{3}{5}}$.

Part (a) Step 1. Write with exponents.

Consider the given expression,

$\sqrt{m^5}$

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, $5$.

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

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

Part (b) Step 1. Write with exponents.

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 ${\left(3y\right)}^{\frac{7}{3}}$.

Part (c) Step 1. Write with exponents.

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 ${\left(\frac{4x}{5y}\right)}^{\frac{3}{5}}$.