Question

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

$$\begin{gathered} \left(a\right){\left(y^3\right)}^x \\ \left(b\right){\left(5^x\right)}^y \\ \left(c\right){\left(q^6\right)}^{-8} \end{gathered}$$

Short answer

The simplified expressions are:

$$\begin{gathered} \left(a\right){\left(y^3\right)}^x=y^{3x} \\ \left(b\right){\left(5^x\right)}^y=5^{xy} \\ \left(c\right){\left(q^6\right)}^{-8}=q^{-48} \end{gathered}$$

Step-by-step solution

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 ${\left(y^3\right)}^x$.

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

Part a Step 2. Simplify

Using the Power property for exponent,

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

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

Part b Step 1. Given information

The given expression is${\left(5^x\right)}^y$.

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

Part b Step 2. Simplify

Using the Product power for exponent,

$\begin{array}{c}{\left(5^x\right)}^y=5^{x\cdot y}\\ =5^{xy}\end{array}$

So, the simplified expression is${\left(5^x\right)}^y=5^{xy}$

Part c Step 1. Given information

The given expression is${\left(q^6\right)}^{-8}$.

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

Part c Step 2. Simplify

Using the Power property for exponent,

$\begin{array}{c}{\left(q^6\right)}^{-8}=q^{6\cdot \left(-8\right)}\\ =q^{-48}\end{array}$

So, the simplified expression is${\left(q^6\right)}^{-8}=q^{-48}$