Q. 143 - Intermediate Algebra

Question

In the following exercises, write with a rational exponent.

$$\begin{gathered} \left(a\right) \sqrt[5]{u^2} \\ \left(b\right) \left(\sqrt[3]{6x}{)}^5 \\ \right(c) \sqrt[4]{{\left(\frac{18a}{5b}\right)}^7} \end{gathered}$$

Step-by-Step Solution

Answer

(a) The answer is $u^{\frac{2}{5}}$

(b) The answer is ${\left(6x\right)}^{\frac{5}{3}}$

(c) The answer is ${\left(\frac{18a}{5b}\right)}^{\frac{7}{4}}$

Step 1. Part (a) Simplify the expression u 2 5

We will use the property 

$a^{\frac{m}{n}}=(\sqrt[n]{a}{)}^m$ and $a^{\frac{m}{n}}=\sqrt[n]{a^m}$

$$\begin{gathered} \sqrt[5]{u^2} \\ ={\left(u^2\right)}^{\frac{1}{5}} \\ =u^{\frac{2}{5}} \end{gathered}$$

Step 2. Part (b) Simplify the expression ( 6 x 3 ) 5

$$\begin{gathered} {\left(\sqrt[3]{6x}\right)}^5 ={\left({\left(6x\right)}^{\frac{1}{3}}\right)}^5 \\ ={\left(6x\right)}^{\frac{5}{3}} \end{gathered}$$

Step 3. Part (c) Simplify the expression 18 a 5 b 7 4

$$\begin{gathered} \sqrt[4]{{\left(\frac{18a}{5b}\right)}^7} \\ ={\left(\frac{18a}{5b}\right)}^{\frac{7}{4}} \end{gathered}$$