Q. 144 - Intermediate Algebra

Question

In the following exercises, write with a rational exponent.

$$\begin{gathered} \left(a\right) \sqrt[3]{a} \\ \left(b\right) \left(\sqrt[4]{21v}{)}^3 \\ \right(c) \sqrt[4]{{\left(\frac{2xy}{5z}\right)}^2} \end{gathered}$$

Step-by-Step Solution

Answer

(a) The answer is $a^{\frac{1}{3}}$

(b) The answer is ${\left(21v\right)}^{\frac{3}{4}}$

(c) The answer is ${\left(\frac{2xy}{5z}\right)}^{\frac{1}{2}}$

Step 1. Part (a) Simplify the expression a 3

We will use the property

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

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

Step 2. Part (b) Simplify the expression 21 v 4 3

$$\begin{gathered} {\left(\sqrt[4]{21v}\right)}^3 \\ ={\left(21v\right)}^{3\times \frac{1}{4}} \\ ={\left(21v\right)}^{\frac{3}{4}} \end{gathered}$$

Step 3. Part (c) Simplify the expression 2 x y 5 z 2 4

$$\begin{gathered} \sqrt[4]{{\left(\frac{2xy}{5z}\right)}^2} \\ ={\left(\frac{2xy}{5z}\right)}^{2\times \frac{1}{4}} \\ ={\left(\frac{2xy}{5z}\right)}^{\frac{1}{2}} \end{gathered}$$