Q. 122. - Intermediate Algebra

Question

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

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

Step-by-Step Solution

Answer

The simplified form of the expressions 

$$\begin{gathered} \left(a\right). {\left(-3xyz\right)}^4=81x^4{y}^4{z}^4 \\ \left(b\right). {\left(-7mn\right)}^0=1 \\ \left(c\right). {\left(-3x^3\right)}^{-2}=\frac{1}{9x^6} \\ \left(d\right). {\left(2y^{-5}\right)}^2=\frac{4}{y^{10}} \end{gathered}$$

Part (a) Step 1. Given information.

The given expression is (-3xyz)⁴.

Part (a) Step 2. Calculation of expression:

In calculation of expression we use concept of product of power-

(ab)ⁿ = aⁿ bⁿ

So,

(-3xyz)⁴ = 81x⁴y⁴z⁴.

Part (b) Sep 1. Given information.

The given expression is (-7mn)⁰.

Part (b) Step 2. Calculation of expression.

Calculate the expression by anything raised to zero is always 1.

So,

(-7mn)⁰ = 1.

Part (c) Step 1. Given information.

The given expression is (-3x³)^-2.

Part (c) Step 2. Calculation of expression.

Calculate the expression by negative exponent and product of power.

product of power-

(ab)ⁿ = aⁿ bⁿ

product of negative exponent-

(ab)^-n = 1/(ab)ⁿ

We get,

(-3x³)^-2 = $\frac{1}{{(-3x^3)}^2}$

             =$\frac{1}{9x^6}$

Part (d) Step 1. Given information.

The given expression is (2y^-5)².

Part (d) Step 2. Calculate the expression.

Apply the concept of power product-

(ab)ⁿ = aⁿ bⁿ

So,

(2y^-5)² = 4y^-10

            =$\frac{4}{y^{10}}$