Q 142E - Intermediate Algebra

Question

In the following exercises, write with a rational exponent.

Part (a): $\sqrt[4]{r^7}$

Part (b): ${\left(\sqrt[5]{2pq}\right)}^3$

Part (c): $\sqrt[4]{{\left(\frac{12m}{7n}\right)}^3}$

Step-by-Step Solution

Answer

Part (a): On writing with a rational exponent is $r^{\frac{7}{4}}$.

Part (b): On writing with a rational exponent is ${\left(2pq\right)}^{\frac{3}{5}}$.

Part (c): On writing with a rational exponent is ${\left(\frac{12m}{7n}\right)}^{\frac{3}{4}}$.

Part (a) Step 1. Write with exponents.

Consider the given expression,

$\sqrt[4]{r^7}$

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, $4$.

Therefore, the answer is $r^{\frac{7}{4}}$.

Part (b) Step 1. Write with exponents.

Consider the given expression,

${\left(\sqrt[5]{2pq}\right)}^3$

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 ${\left(2pq\right)}^{\frac{3}{5}}$.

Part (c) Step 1. Write with exponents.

Consider the given expression,

$\sqrt[4]{{\left(\frac{12m}{7n}\right)}^3}$

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, $4$.

Therefore, the answer is ${\left(\frac{12m}{7n}\right)}^{\frac{3}{4}}$.