12.8: Recognize and Apply the Power of a Product Property
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*}
\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*}
\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*}
(ab)n=an(bn)
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*}
(6x5)2 
\begin{align*}(13d^5)^2\end{align*}
(−13d5)2 
\begin{align*}(3p^3 q^4)^3\end{align*}
(−3p3q4)3 
\begin{align*}(10xy^2)^4\end{align*}
(10xy2)4 
\begin{align*}(4t^3)^5\end{align*}
(−4t3)5 
\begin{align*}(18 r^2 s^3)^2\end{align*}
(18r2s3)2 
\begin{align*}(2r^{11}s^3 t^2)^3\end{align*}
(2r11s3t2)3 
\begin{align*}(7x^2)^2\end{align*}
(7x2)2 
\begin{align*}(2y^2)^3\end{align*}
(2y2)3 
\begin{align*}(5x^2)^3\end{align*}
(5x2)3 
\begin{align*}(12y^3)^2\end{align*}
(12y3)2 
\begin{align*}(5x^5)^5\end{align*}
(5x5)5 
\begin{align*}(2x^2y^2z)^3\end{align*}
(2x2y2z)3 
\begin{align*}(3x^4y^3z^2)^3\end{align*}
(3x4y3z2)3 
\begin{align*}(5x^4y^3z^3)^3\end{align*}
(−5x4y3z3)3
Notes/Highlights Having trouble? Report an issue.
Color  Highlighted Text  Notes  

Please Sign In to create your own Highlights / Notes  
Show More 
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.