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# Exponential Properties Involving Quotients

## Subtract exponents to divide exponents by other exponents

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Practice Exponential Properties Involving Quotients
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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 12121212\begin{align*}\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}\end{align*} or (12)4\begin{align*}\left(\frac{1}{2} \right)^4\end{align*}. We can use exponents with fractions or quotients, too. In order to answer the question above, we would multiply the numerators and denominators across, like this: 11112222=116\begin{align*}\frac{1 \cdot 1 \cdot 1 \cdot 1}{2 \cdot 2 \cdot 2 \cdot 2}=\frac{1}{16}\end{align*}. Half of a half of a half of a half is one sixteenth. Once again, we have repeating multiplication of the same number which we could write more easily as 1424=116\begin{align*}\frac{1^4}{2^4}=\frac{1}{16}\end{align*}.

The Power of a Quotient Property says that 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

Here is one for you to try.

(53)4=5434=62581\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.

(3k2j)4=(3k)4(2j)4=(3k)(3k)(3k)(3k)(2j)(2j)(2j)(2j)=81k416j4\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

(45)3\begin{align*}\left(\frac{4}{5}\right)^3 \end{align*}

Solution: 64125=.512\begin{align*}\frac{64}{125} = .512\end{align*}

#### Example B

(2a3b)2\begin{align*}\left(\frac{2a}{3b}\right)^2 \end{align*}

Solution: 4a29b2\begin{align*}\frac{4a^2}{9b^2}\end{align*}

#### Example C

(a5b)3\begin{align*}\left(\frac{a}{5b}\right)^3 \end{align*}

Solution: a3125b3\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 13\begin{align*}\frac{1}{3}\end{align*} to the fourth power.

Here is how we can set up the problem.

10(13)4=10(1434)=10(181)=1081 grams\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*}
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*}

\begin{align*}\frac{-64x^3}{27y^3}\end{align*}

### Practice

Directions: Simplify.

1. \begin{align*}\left(\frac{2}{3}\right)^4 \end{align*}
2. \begin{align*}\left(\frac{1}{3}\right)^3 \end{align*}
3. \begin{align*}\left(\frac{7}{8}\right)^2 \end{align*}
4. \begin{align*}\left(\frac{2}{5}\right)^4 \end{align*}
5. \begin{align*}\left(\frac{7k}{-2m}\right)^3\end{align*}
6. \begin{align*}\left(\frac{3x}{-2y}\right)^3\end{align*}
7. \begin{align*}\left(\frac{4x}{-3y}\right)^4\end{align*}
8. \begin{align*}\left(\frac{5y}{-2z}\right)^5\end{align*}
9. \begin{align*}\left(\frac{-2y}{4z}\right)^4\end{align*}
10. \begin{align*}\left(\frac{4xy}{-2z^5}\right)^5\end{align*}
11. \begin{align*}\left(\frac{12x^2y^4}{-6z^3}\right)^2\end{align*}
12. \begin{align*}\left(\frac{7x^2y}{-2z^3}\right)^3\end{align*}
13. \begin{align*}\left(\frac{2x^3y^2}{-2z^3}\right)^3\end{align*}
14. \begin{align*}\left(\frac{x^{11}}{y^9}\right)^5\end{align*}
15. \begin{align*}\left(\frac{-5x^3}{3h^2 j^8}\right)^5\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".
Power of a Product Property

Power of a Product Property

The power of a product property states that $(ab)^m = a^m b^m$.
Power of a Quotient Property

Power of a Quotient Property

The power of a quotient property states that $\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}$.
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.