12.9: Recognize and Apply the Power of a Quotient Property
Have you ever been in a science laboratory? Take a look at this dilemma.
One of the places that the students were able to visit when they went downtown was a laboratory at the city college. Downtown, the city college had some of its classrooms and one of the classrooms was a laboratory.
“This is a good friend of mine Professor Smith,” Mr. Travis said introducing the students to a woman with blonde hair and a wide smile.
“Welcome,” Professor Smith said. “Are you enjoying your trip downtown?”
Many students responded yes and then were drawn over to one of the laboratory tables where a lot of work was taking place.
“What is happening here?” Sam asked.
“Well, I started with a very small sample of cobalt. I actually had 10 grams of it and I took a third of a third of a third of a third of it,” She explained.
The students began figuring the math out in their heads.
Can you figure it out? How many grams did the sample end up being? By the end of this Concept, you will be able to solve this dilemma.
Guidance
This may sound confusing, but in math, we can rewrite this as \begin{align*}\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}\end{align*}
The Power of a Quotient Property says that for any nonzero numbers \begin{align*}a\end{align*}
\begin{align*}\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}\end{align*}
Here is one for you to try.
\begin{align*}\left(\frac{5}{3}\right)^4=\frac{5^4}{3^4}=\frac{625}{81}\end{align*}
You can see in this situation that we have simplified the expression by figuring out what five to the fourth is and what three to the fourth is. The next step in this problem would be to divide.
Take a look at this one.
\begin{align*}\left(\frac{3k}{2j}\right)^4=\frac{(3k)^4}{(2j)^4}=\frac{(3k)(3k)(3k)(3k)}{(2j)(2j)(2j)(2j)}=\frac{81k^4}{16j^4}\end{align*}
This problem has different variables, so this is as far as we can take this problem.
Simply each quotient.
Example A
\begin{align*}\left(\frac{4}{5}\right)^3 \end{align*}
Solution: \begin{align*}\frac{64}{125} = .512\end{align*}
Example B
\begin{align*}\left(\frac{2a}{3b}\right)^2 \end{align*}
Solution: \begin{align*}\frac{4a^2}{9b^2}\end{align*}
Example C
\begin{align*}\left(\frac{a}{5b}\right)^3 \end{align*}
Solution: \begin{align*}\frac{a^3}{125b^3}\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
To figure out the number of grams in the sample, we must use what we have learned about monomials and powers.
Professor Smith started off with 10 grams.
Then she took a third of a third of a third of a third of it. That is \begin{align*}\frac{1}{3}\end{align*}
Here is how we can set up the problem.
\begin{align*}10 \left(\frac{1}{3}\right)^4=10 \left(\frac{1^4}{3^4}\right)=10 \left (\frac{1}{81}\right)=\frac{10}{81} \ grams\end{align*}
We can convert that into a decimal by dividing the numerator by the denominator.
.12 grams is our answer as a decimal.
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)
 Power of a Quotient Property
 the exponent is applied to both the top and bottom numbers in an expression.
Guided Practice
Here is one for you to try on your own.
Simplify the following quotient.
\begin{align*}\left(\frac{4x}{3y}\right)^3 \end{align*}
Solution
First, let's work with the numerator.
\begin{align*}(4x)^3 = 64x^3\end{align*}
Now let's work with the denominator.
\begin{align*}(3y)^3 = 27y^3\end{align*}
Here is our final answer.
\begin{align*}\frac{64x^3}{27y^3}\end{align*}
Video Review
Multiplying and Dividing Monomials
Practice
Directions: Simplify.

\begin{align*}\left(\frac{2}{3}\right)^4 \end{align*}
(23)4 
\begin{align*}\left(\frac{1}{3}\right)^3 \end{align*}
(13)3 
\begin{align*}\left(\frac{7}{8}\right)^2 \end{align*}
(78)2 
\begin{align*}\left(\frac{2}{5}\right)^4 \end{align*}
(25)4 
\begin{align*}\left(\frac{7k}{2m}\right)^3\end{align*}
(7k−2m)3 
\begin{align*}\left(\frac{3x}{2y}\right)^3\end{align*}
(3x−2y)3 
\begin{align*}\left(\frac{4x}{3y}\right)^4\end{align*}
(4x−3y)4 
\begin{align*}\left(\frac{5y}{2z}\right)^5\end{align*}
(5y−2z)5  \begin{align*}\left(\frac{2y}{4z}\right)^4\end{align*}
 \begin{align*}\left(\frac{4xy}{2z^5}\right)^5\end{align*}
 \begin{align*}\left(\frac{12x^2y^4}{6z^3}\right)^2\end{align*}
 \begin{align*}\left(\frac{7x^2y}{2z^3}\right)^3\end{align*}
 \begin{align*}\left(\frac{2x^3y^2}{2z^3}\right)^3\end{align*}
 \begin{align*}\left(\frac{x^{11}}{y^9}\right)^5\end{align*}
 \begin{align*}\left(\frac{5x^3}{3h^2 j^8}\right)^5\end{align*}
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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".Power of a Product Property
The power of a product property states that .Power of a Quotient Property
The power of a quotient 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 quotient property.