Q. 152 - Intermediate Algebra

Question

In the following exercises, simplify.

(a) $6^{\frac{5}{2}}·6^{\frac{1}{2}}$

(b) ${\left(b^{15}\right)}^{\frac{3}{5}}$

(c) $\frac{w^{{\displaystyle \frac{2}{7}}}}{w^{{\displaystyle \frac{9}{7}}}}$

Step-by-Step Solution

Answer

(a) The answer is $216$

(b) The answer is $b^9$

(c) The answer is $\frac{1}{w}$

Step 1. Part (a) Simplify the expression 6 5 2 · 6 1 2

We will use the property

$a^m·a^n=a^{m+n}$

$$\begin{gathered} 6^{\frac{5}{2}}·6^{\frac{1}{2}} \\ =6^{\frac{5}{2}+\frac{1}{2}} \\ =6^{\frac{6}{2}} \\ =6^3 \\ =216 \end{gathered}$$

Step 2. Part (b) Simplify the expression b 15 3 5

We will use the property

${\left(a^m\right)}^n=a^{mn}$

$$\begin{gathered} {\left(b^{15}\right)}^{\frac{3}{5}} \\ =b^{15\times \frac{3}{5}} \\ =b^9 \end{gathered}$$

Step 3. Part (c) Simplify the expression w 2 7 w 9 7

We will use the property

$\frac{a^m}{a^n}=a^{m-n}$

$$\begin{gathered} \frac{w^{{\displaystyle \frac{2}{7}}}}{w^{{\displaystyle \frac{9}{7}}}} \\ =w^{\frac{2}{7}-\frac{9}{7}} \\ =w^{-\frac{7}{7}} \\ =w^{-1} \\ =\frac{1}{w} \end{gathered}$$