# 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*}

### Notes/Highlights Having trouble? Report an issue.

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Term | Definition |
---|---|

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.