Q. 118 - Intermediate Algebra

Question

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

$$\begin{gathered} \left(a\right) {\left(x^2\right)}^y \\ \left(b\right) {\left(7^a\right)}^b \\ \left(c\right) {\left(a^9\right)}^{-10} \end{gathered}$$

Step-by-Step Solution

Answer

The simplified expression are:

$$\begin{gathered} \left(a\right) {\left(x^2\right)}^y=x^{2y} \\ \left(b\right) {\left(7^a\right)}^b=7^{ab} \\ \left(c\right) {\left(a^9\right)}^{-10}=a^{-90} \end{gathered}$$

Step 1. Power properties 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 ${\left(x^2\right)}^y$.

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

Part a Step 2. Simplify

Using Power property for exponent,

$\begin{array}{c}{\left(x^2\right)}^y=x^{2\cdot y}\\ =x^{2y}\end{array}$

So, the simplified expression is ${\left(x^2\right)}^y=x^{2y}$

Part b Step 1. Given information

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

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

Part b Step 2. Simplify

Using the Power property for exponents,

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

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

Part c Step 1. Given information

The given expression is ${\left(a^9\right)}^{-10}$ .

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

Part c Step 2. Simplify

Using Power properties for exponent,

$\begin{array}{c}{\left(a^9\right)}^{-10}=a^{9\cdot \left(-10\right)}\\ =a^{-90}\end{array}$

So, the simplified expression is ${\left(a^9\right)}^{-10}=a^{-90}$