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

Difficulty Level: At Grade Created by: CK-12
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Practice Exponential Properties Involving Products
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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 6a2\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.

(5x)4\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 x\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 4th\begin{align*}4^{th}\end{align*} power. We can also write this in expanded form.

(5x)4=(5x)(5x)(5x)(5x)

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.

=5555xxxx=625x4

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

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

Here is another one.

(7h)3=(7h)(7h)(7h)=777hhh=343h3

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:

aman=am+n

Take a look at this one.

(2x4)5=(2x4)(2x4)(2x4)(2x4)(2x4)=22222x4x4x4x4x4=22222x4+4+4+4+4=32x20

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

Simplify each monomial.

#### Example A

(6x3)2\begin{align*}(6x^3)^2\end{align*}

Solution: 36x6\begin{align*}36x^6\end{align*}

#### Example B

(2x3y3)3\begin{align*}(2x^3y^3)^3\end{align*}

Solution: 8x9y9\begin{align*}8x^9y^9\end{align*}

#### Example C

(3x2y2z)4\begin{align*}(-3x^2y^2z)^4\end{align*}

Solution: 81x8y8z4\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.

6a2\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.

A=s2\begin{align*}A = s^2\end{align*}

Now we substitute the side length into the formula.

A=(6a2)2\begin{align*}A = (6a^2)^2\end{align*}

Next, we can square the monomial.

36a4\begin{align*}36a^4\end{align*}

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

### Guided Practice

Here is one for you to try on your own.

(2x4)5\begin{align*}(-2x^4)^5\end{align*}

Solution

(2x4)5=(2x4)(2x4)(2x4)(2x4)(2x4)=(2xxxx)(2xxxx)(2xxxx)(2xxxx)=22222xxxxxxxxxxxxxxxxxxxx=32x20

### Practice

Directions: Simplify.

1. (6x5)2\begin{align*}(6x^5)^2\end{align*}
2. (13d5)2\begin{align*}(-13d^5)^2\end{align*}
3. (3p3q4)3\begin{align*}(-3p^3 q^4)^3\end{align*}
4. (10xy2)4\begin{align*}(10xy^2)^4\end{align*}
5. (4t3)5\begin{align*}(-4t^3)^5\end{align*}
6. (18r2s3)2\begin{align*}(18 r^2 s^3)^2\end{align*}
7. (2r11s3t2)3\begin{align*}(2r^{11}s^3 t^2)^3\end{align*}
8. (7x2)2\begin{align*}(7x^2)^2\end{align*}
9. (2y2)3\begin{align*}(2y^2)^3\end{align*}
10. (5x2)3\begin{align*}(5x^2)^3\end{align*}
11. (12y3)2\begin{align*}(12y^3)^2\end{align*}
12. (5x5)5\begin{align*}(5x^5)^5\end{align*}
13. (2x2y2z)3\begin{align*}(2x^2y^2z)^3\end{align*}
14. (3x4y3z2)3\begin{align*}(3x^4y^3z^2)^3\end{align*}
15. (5x4y3z3)3\begin{align*}(-5x^4y^3z^3)^3\end{align*}

### Vocabulary Language: English

Base

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 $32^4$, 32 is the base, and 4 is the exponent.
Coefficient

Coefficient

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

Expanded Form

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

Exponent

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

Monomial

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

Power

The "power" refers to the value of the exponent. For example, $3^4$ is "three to the fourth power".
Product of Powers Property

Product of Powers Property

The product of powers property states that $a^m \cdot a^n = a^{m+n}$.
Variable

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.

Mar 19, 2013

Nov 20, 2015