Q. 115 - Intermediate Algebra

Question

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

$\begin{array}{l}\left(a\right) {\left(m^4\right)}^2\\ \left(b\right) {\left({10}^3\right)}^6\\ \left(c\right) {\left(x^3\right)}^{-4}\end{array}$

Step-by-Step Solution

Answer

The simplified expression are:

$\begin{array}{l}\left(a\right) {\left(m^4\right)}^2= m^8\\ \left(b\right) {\left({10}^3\right)}^6= {10}^{18}\\ \left(c\right) {\left(x^3\right)}^{-4}= x^{-12}\end{array}$

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 ${\left(m^4\right)}^2$.

We have to simplify the expression using Power property for exponents.

Part a Step 2. Simplify

Using Power property for exponents,

$\begin{array}{c}{\left(m^4\right)}^2=m^{4\cdot 2}\\ =m^8\end{array}$

So, The simplified expression is ${\left(m^4\right)}^2= m^8$

Part b Step 1. Given information

The given expression is ${\left({10}^3\right)}^6$

We have to simplify the expression using Power property for exponents.

Part b Step 2. Simplify

Using Power property for exponents,

$\begin{array}{c}{\left({10}^3\right)}^6={10}^{3\cdot 6}\\ ={10}^{18}\end{array}$

So, the simplified expression is ${\left({10}^3\right)}^6={10}^{18}$

Part c Step 1. Given information

The given expression is ${\left(x^3\right)}^{-4}$.

We have to simplify the expression using Power property for exponents.

Part c Step 2. Simplify

Using Power property for exponents,

$\begin{array}{c}{\left(x^3\right)}^{-4}=x^{3\cdot \left(-4\right)}\\ =x^{-12}\end{array}$

So, the simplified expression is ${\left(x^3\right)}^{-4}= x^{-12}$