Question

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

$$\begin{gathered} \left(a\right).{\left(-5ab\right)}^3 \\ \left(b\right).{\left(-4pq\right)}^0 \\ \left(c\right).{\left(-6x^3\right)}^{-2} \\ \left(d\right).{\left(3y^{-4}\right)}^2 \end{gathered}$$

Short answer

The simplified form is

$$\begin{gathered} \left(a\right).{\left(-5ab\right)}^3=-125a^3{b}^3 \\ \left(b\right).\left(-4pq\right)0=1 \\ \left(c\right).{\left(-6x^3\right)}^{-2}=\frac{1}{36x^6} \\ \left(d\right).{\left(3y^{-4}\right)}^2=\frac{9}{y^8} \end{gathered}$$

Step-by-step solution

Part (a) Step 1. The given expression is -5ab3

Given, ${\left(-5ab\right)}^3$

Now apply product to a power property

We get,

$$\begin{gathered} {\left(-5\right)}^3·a^3·b^3 \\ =-125a^3{b}^3 \end{gathered}$$

Part (b) Step 1. The given expression is -4pq0

Given, ${\left(-4pq\right)}^0$

Zero exponent property $a^0=1$

We get,

${\left(-4pq\right)}^0=1$

Part (c) Step 1. The given expression is -6x3-2

Given, ${\left(-6x^3\right)}^{-2}$

First of all, apply product to a power property

We get,

${\left(-6\right)}^{-2}·{\left(x^3\right)}^{-2}$

Now, apply properties of Negative exponents $a^{-n}=\frac{1}{n}$

We get,

$$\begin{gathered} \frac{1}{{\left(-6\right)}^2}·\frac{1}{{\left(x^3\right)}^2} \\ =\frac{1}{36}·\frac{1}{x^6} \\ =\frac{1}{36x^6} \end{gathered}$$

Part (d) Step 1. The given expression is 3y-42

Given, ${\left(3y^{-4}\right)}^2$

First of all, apply product to a power property

We get,

$$\begin{gathered} 3^2·{\left(y^{-4}\right)}^2 \\ =9·{\left(y^{-4}\right)}^2 \end{gathered}$$

Now, apply power property${\left(a^m\right)}^n=a^{m·n}$

We get,

$$\begin{gathered} 9·{\left(y\right)}^{-8} \\ =9·\frac{1}{y^8} \\ =\frac{9}{y^8} \end{gathered}$$