# 12.8: Recognize and Apply the Power of a Product Property

**At Grade**Created by: CK-12

**Practice**Exponential Properties Involving Products

Have you ever tried to square a monomial? Do you know how to do it? Take a look at this dilemma.

A square platform has a side length of \begin{align*}6a^2\end{align*}.

How can we find the area of the platform?

**This Concept will show you how to use the Power of a Product with monomials. Then you will be able to find the area of the square platform.**

### Guidance

When multiplying monomials, an exponent is applied to the constant, variable, or quantity that is directly to its left. However, we only applied exponents to single variables.

**Exponents can also be applied to products using parentheses.**

Look at this one.

**\begin{align*}(5x)^4\end{align*}**

**If we apply the exponent 4 to whatever is directly to its left, we would apply it to the parentheses, not just the \begin{align*}x\end{align*}. The parentheses are directly to the left of the 4. This indicates that the entire product in the parentheses is taken to the \begin{align*}4^{th}\end{align*} power. We can also write this in expanded form.**

\begin{align*}& (5x)^4 \\ &=(5x)(5x)(5x)(5x)\end{align*}

**Now we multiply the monomials as we have already learned—by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.**

\begin{align*}&=5 \cdot 5 \cdot 5 \cdot 5 \cdot x \cdot x \cdot x \cdot x \\ &=625x^4\end{align*}

**This is the** *Power of a Product Property***which says, for any nonzero numbers \begin{align*}a\end{align*} and \begin{align*}b\end{align*} and any integer \begin{align*}n\end{align*}**

\begin{align*}(ab)^n=a^n b^n\end{align*}

Here is another one.

\begin{align*} & (7h)^3 \\ &=(7h)(7h)(7h) \\ &=7 \cdot 7 \cdot 7 \cdot h \cdot h \cdot h \\ &=343 h^3 \end{align*}

You can see that whether we have positive or negative integers or both, we can still use the Power of a Product Property. You may have already noticed a pattern with the exponents and the final product. When you multiply like bases, there is another shortcut—you can add the exponents of like bases. Another way of saying it is:

\begin{align*}a^m \cdot a^n=a^{m+n}\end{align*}

Take a look at this one.

\begin{align*}& (-2x^4)^5 \\ &=(-2x^4)(-2x^4)(-2x^4)(-2x^4)(-2x^4) \\ &=-2 \cdot -2 \cdot -2 \cdot -2 \cdot -2 \cdot x^4 \cdot x^4 \cdot x^4 \cdot x^4 \cdot x^4 \\ &=-2 \cdot -2 \cdot -2 \cdot -2 \cdot -2 \cdot x^{4+4+4+4+4}\\ &=-32x^{20} \end{align*}

*Write the definition of this property and one problem down in your notebook.*

Simplify each monomial.

#### Example A

\begin{align*}(6x^3)^2\end{align*}

**Solution: \begin{align*}36x^6\end{align*}**

#### Example B

\begin{align*}(2x^3y^3)^3\end{align*}

**Solution: \begin{align*}8x^9y^9\end{align*}**

#### Example C

\begin{align*}(-3x^2y^2z)^4\end{align*}

**Solution: \begin{align*}81x^8y^8z^4\end{align*}**

Now let's go back to the dilemma from the beginning of the Concept.

Here is the side length of the square platform.

\begin{align*}6a^2\end{align*}

We want to find the area of the platform. To figure out the area, we will use the following formula.

\begin{align*}A = s^2\end{align*}

Now we substitute the side length into the formula.

\begin{align*}A = (6a^2)^2\end{align*}

Next, we can square the monomial.

\begin{align*}36a^4\end{align*}

**This is our answer.**

### Vocabulary

- Monomial
- a single term of variables, coefficients and powers.

- Coefficient
- the number part of a monomial or term.

- Variable
- the letter part of a term

- Exponent
- the little number, the power, that tells you how many times to multiply the base by itself.

- Base
- the number that is impacted by the exponent.

- Expanded Form
- write out all of the multiplication without an exponent.

- Power of a Product Property
- \begin{align*}(ab)^n=a^n(b^n)\end{align*}

### Guided Practice

Here is one for you to try on your own.

\begin{align*}(-2x^4)^5\end{align*}

**Solution**

\begin{align*}& (-2x^4)^5 \\ &=(-2x^4)(-2x^4)(-2x^4)(-2x^4)(-2x^4) \\ &=(-2 \cdot x \cdot x \cdot x \cdot x)(-2 \cdot x \cdot x \cdot x \cdot x)(-2 \cdot x \cdot x \cdot x \cdot x)(-2 \cdot x \cdot x \cdot x \cdot x) \\ &=-2 \cdot -2 \cdot -2 \cdot -2 \cdot -2 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \\ &=-32x^{20} \end{align*}

### Video Review

### Practice

Directions: Simplify.

- \begin{align*}(6x^5)^2\end{align*}
- \begin{align*}(-13d^5)^2\end{align*}
- \begin{align*}(-3p^3 q^4)^3\end{align*}
- \begin{align*}(10xy^2)^4\end{align*}
- \begin{align*}(-4t^3)^5\end{align*}
- \begin{align*}(18 r^2 s^3)^2\end{align*}
- \begin{align*}(2r^{11}s^3 t^2)^3\end{align*}
- \begin{align*}(7x^2)^2\end{align*}
- \begin{align*}(2y^2)^3\end{align*}
- \begin{align*}(5x^2)^3\end{align*}
- \begin{align*}(12y^3)^2\end{align*}
- \begin{align*}(5x^5)^5\end{align*}
- \begin{align*}(2x^2y^2z)^3\end{align*}
- \begin{align*}(3x^4y^3z^2)^3\end{align*}
- \begin{align*}(-5x^4y^3z^3)^3\end{align*}

Base

When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression , 32 is the base, and 4 is the exponent.Coefficient

A coefficient is the number in front of a variable.Expanded Form

Expanded form refers to a base and an exponent written as repeated multiplication.Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.Monomial

A monomial is an expression made up of only one term.Power

The "power" refers to the value of the exponent. For example, is "three to the fourth power".Product of Powers Property

The product of powers property states that .Variable

A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.### Image Attributions

Here you'll recognize and apply the power of a product property when multiplying monomials.

## Concept Nodes:

Base

When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression , 32 is the base, and 4 is the exponent.Coefficient

A coefficient is the number in front of a variable.Expanded Form

Expanded form refers to a base and an exponent written as repeated multiplication.Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.Monomial

A monomial is an expression made up of only one term.Power

The "power" refers to the value of the exponent. For example, is "three to the fourth power".Product of Powers Property

The product of powers property states that .Variable

A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.