Question

In the following exercises, simplify each expression using the properties for exponents.

$$\begin{gathered} \left(a\right)\frac{p^{21}}{p^7} \\ \left(b\right)\frac{4^{16}}{4^4} \\ \left(c\right)\frac{b}{b^9} \\ \left(d\right)\frac{4}{4^6} \end{gathered}$$

Short answer

The simplified form is:

$\begin{array}{rcl}\left(a\right)\frac{p^{21}}{p^7}& =& p^{14}\\ \left(b\right)\frac{4^{16}}{4^4}& =& 4^{12}\\ \left(c\right)\frac{b}{b^9}& =& \frac{1}{b^8}\\ \left(d\right)\frac{4}{4^6}& =& \frac{1}{4^5}\end{array}$

Step-by-step solution

Part(a)-Step 1. Given information

The given information is$\frac{p^{21}}{p^7}$.

The main objective is to simplify the expression.

Part(a)-Step 2. Simplify

Use quotient property $\frac{a^m}{a^n}=a^{m-n},m>n$to simplify the expressions:

$\begin{array}{rcl}\frac{p^{21}}{p^7}& =& p^{21-7}\\ & =& p^{14}\end{array}$.

Part(b)-Step 1. Given information.

The given information is$\frac{4^{16}}{4^4}$.

The main objective is to simplify the expression.

Part(b)-Step2. Simplify

Use the property that $\frac{a^m}{a^n}=a^{m-n},m>n$.

$\begin{array}{rcl}\frac{4^{16}}{4^4}& =& 4^{16-4}\\ & =& 4^{12}\end{array}$

Part(c)-Step 1. Given information

The given information is$\frac{b}{b^9}$.

The main objective is to simplify the expression.

Part(c)-Step 2. Simplify

Use the property that$\frac{a^m}{a^n}=\frac{1}{a^{n-m}},n>m$.

$\begin{array}{rcl}\frac{b}{b^9}& =& \frac{1}{b^{9-1}}\\ & =& \frac{1}{b^8}\end{array}$

Part(d)-Step 1. Given information

The given information is$\frac{4}{4^6}$.

The main objective is to simplify the expression.

Part(d)-Step 2. Simplify

Use the property that$\frac{a^m}{a^n}=\frac{1}{a^{n-m}},n>m$.

$\begin{array}{rcl}\frac{4}{4^6}& =& \frac{4^1}{4^6}\\ & =& \frac{1}{4^{6-1}}\\ & =& \frac{1}{4^5}\end{array}$