Question

In the following exercises, write with a rational exponent.

Part (a):$\sqrt[3]{25a}$

Part (b): $\sqrt{3b}$

Part (c):$\sqrt[8]{40c}$

Short answer

Part (a): On writing with a rational exponent form, we get${\left(25a\right)}^{\frac{1}{3}}$.

Part (b): On writing with a rational exponent form, we get localid="1663339405840" ${\left(3b\right)}^{\frac{1}{2}}$.

Part (c): On writing with a rational exponent form, we get ${\left(40c\right)}^{\frac{1}{8}}$.

Step-by-step solution

Part (a) Step 1. Write with a rational exponent.

Consider the given question,

$\sqrt[3]{25a}$

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

Therefore, we can write,

$={\left(25a\right)}^{\frac{1}{3}}$

Part (b) Step 1. Write with a rational exponent.

Consider the given question,

$\sqrt{3b}$

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

Therefore, we can write,

localid="1663339552614" $={\left(3b\right)}^{\frac{1}{2}}$

Part (c) Step 1. Write with a rational exponent.

Consider the given question,

$\sqrt[8]{40c}$

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

Therefore, we can write,

$={\left(40c\right)}^{\frac{1}{8}}$