Q. 151 - Intermediate Algebra

Question

In the following exercises, simplify.

(a) $c^{\frac{1}{4}}·c^{\frac{5}{8}}$

(b) ${\left(p^{12}\right)}^{\frac{3}{4}}$

(c) $\frac{r^{{\displaystyle \frac{4}{5}}}}{r^{{\displaystyle \frac{9}{5}}}}$

Step-by-Step Solution

Answer

(a) The answer is $c^{\frac{7}{8}}$

(b) The answer is $p^9$

(c) The answer is $\frac{1}{r}$

Step 1. Part (a) Simplify the expression c 1 4 · c 5 8

We will use the property

$a^m·a^n=a^{m+n}$

$$\begin{gathered} c^{\frac{1}{4}}·c^{\frac{5}{8}} \\ =c^{\frac{1}{4}+\frac{5}{8}} \\ =c^{\frac{2+5}{8}} \\ =c^{\frac{7}{8}} \end{gathered}$$

Step 2. Part (b) Simplify the expression p 12 3 4

We will use the property

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

$$\begin{gathered} {\left(p^{12}\right)}^{\frac{3}{4}} \\ =p^{12\times \frac{3}{4}} \\ =p^9 \end{gathered}$$

Step 3. Part (c) Simplify the expression r 4 5 r 9 5

We will use the property

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

$$\begin{gathered} \frac{r^{{\displaystyle \frac{4}{5}}}}{r^{{\displaystyle \frac{9}{5}}}} \\ =r^{\frac{4}{5}-\frac{9}{5}} \\ =r^{-1} \\ =\frac{1}{r} \end{gathered}$$