Question

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

$$\begin{gathered} \left(a\right).{\left(-4ab\right)}^2 \\ \left(b\right).{\left(5x\right)}^0 \\ \left(c\right).{\left(4y^3\right)}^{-3} \\ \left(d\right).{\left(-7y^{-3}\right)}^2 \end{gathered}$$

Short answer

The simplified form of the expressions

$$\begin{gathered} \left(a\right).{\left(-4ab\right)}^2=16a^2{b}^2 \\ \left(b\right).{\left(5x\right)}^0=1 \\ \left(c\right).{\left(4y^3\right)}^{-3}=\frac{1}{64y^9} \\ \left(d\right).{\left(-7y^{-3}\right)}^2=\frac{49}{y^6} \end{gathered}$$

Step-by-step solution

a. The given expression is -4ab2

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

Now, apply product to a power property

We get,

$$\begin{gathered} {\left(-4\right)}^2·a^2·b^2 \\ =16a^2{b}^2 \end{gathered}$$

b. The given expression is 5x0

Given, ${\left(5x\right)}^0$

Now, apply zero exponent property

We get,

${\left(5x\right)}^0=1$

c. The given expression is 4y3-3

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

Apply product to the power property

We get,

${\left(4\right)}^{-3}·{\left(y^3\right)}^{-3}$

Now, apply properties of negative exponents

We get,

$\frac{1}{4^3}·\frac{1}{{\left(y^3\right)}^3}$

Now, apply power property

We get,

$$\begin{gathered} \frac{1}{64}·\frac{1}{y^9} \\ =\frac{1}{64y^9} \end{gathered}$$

d. The given expression is -7y-32

Given, ${\left(-7y^{-3}\right)}^2$

First of all, apply product to a power property

We get,

${\left(-7\right)}^2·{\left(y^{-3}\right)}^2$

Now, apply power property and property of negative exponents

We get,

$$\begin{gathered} 49·{\left(y\right)}^{-6} \\ =49·\frac{1}{y^6} \\ =\frac{49}{y^6} \end{gathered}$$