Q. 116 - Intermediate Algebra

Question

In the following exercises, simplify each expression using the Power Property. 

$$\begin{gathered} \left(a\right) {\left(b^2\right)}^7 \\ \left(b\right) {\left(3^8\right)}^2 \\ \left(c\right) {\left(k^2\right)}^{-5} \end{gathered}$$

Step-by-Step Solution

Answer

The simplified expression are:

$$\begin{gathered} \left(a\right) {\left(b^2\right)}^7=b^{14} \\ \left(b\right) {\left(3^8\right)}^2=3^{16} \\ \left(c\right) {\left(k^2\right)}^{-5}=k^{-10} \end{gathered}$$

Step 1. Power property for exponents

If a is a real number and m and n are integers, then 

${\left(a^m\right)}^n=a^{m·n}$

Part a Step 1. Given information

The given expression is ${\left(b^2\right)}^7$.

We have to simplify the given expression using Power property for exponent.

Part a Step 2. Simplify

Using the Power property of exponents.

$\begin{array}{c}{\left(b^2\right)}^7=b^{2\cdot 7}\\ =b^{14}\end{array}$

So, the simplified expression is  ${\left(b^2\right)}^7=b^{14}$

Part b Step 1. Given information

The given expression is ${\left(3^8\right)}^2$.

We have to simplify the given expression using Power property for exponent.

Part b Step 2. Simplify

Using the Power property for exponent,

$\begin{array}{c}{\left(3^8\right)}^2=3^{8\cdot 2}\\ =3^{16}\end{array}$

So, the simplified expression is ${\left(3^8\right)}^2=3^{16}$

Part c Step 1. Given information

The given expression is ${\left(k^2\right)}^{-5}$.

We have to simplify the given expression using Power property for exponent.

Part c Step 2. Simplify

Using the Power property for exponents

$\begin{array}{c}{\left(k^2\right)}^{-5}=k^{2\cdot \left(-5\right)}\\ =k^{-10}\end{array}$

So, the simplified the expression is ${\left(k^2\right)}^{-5}=k^{-10}$