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Exponential Properties Involving Quotients

Subtract exponents to divide exponents by other exponents

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Recognize and Apply the Power of a Quotient Property

Let’s Think About It

Credit: US Army RDECOM
Source: https://www.flickr.com/photos/rdecom/6990227636/in/photolist-bDGK99-7JibcX-6Grete-8MTtZL-8MTYtL-bSBnQD-bSBuFa-bDGHdh-d6kmBG-hyxbJ9-bjpM7g-fJGKKh-5VB88j-bDGNJJ-bSBtJB-bDGDMY-7EmMqb-bSByVr-8MRc4X-drtRBb-jKHJcN-8TBEt6-cm9ik9-dUaEcc-f3KEVv-bSBzXF-bDGPV9-bDGJbf-bDGGEU-bSBq4p-bDGEWm-bDGMKJ-drtPAq-8MTV7Q-drtRuE-drtFbx-drtRH1-drXJFQ-drXKfj-drXKc9-6hTqbD-ofdHjA-7BTM1f-azAu8x-8csfmL-78xtJJ-drXK35-drXJRQ-drXJM1-drXJAo
License: CC BY-NC 3.0

Professor Smith works in a laboratory and is training a new intern, Rakesh, in a specific task. Professor Smith tells Rakesh that when she herself did the task, she started with a very small sample of cobalt. She had 10 grams of it and took one-third of one-third of one-third of one-third of it. How can Rakesh figure out how many grams of the sample Professor Smith actually ended up using?

In this concept, you will learn to recognize and apply the power of a quotient property.

Guidance

Exponents can be applied to both fractions and quotients. For example, \begin{align*}\left( \frac{1}{2} \right) \left( \frac{1}{2} \right) \left( \frac{1}{2} \right) \left( \frac{1}{2} \right) = \left( \frac{1}{2} \right)^4 \end{align*}. To evaluate this multiplication, find the product of the numerators and the product of the denominators.

\begin{align*}\begin{array}{rcl} \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} &=& \left( \frac{1}{2} \right)^4 \\ &=& \frac{1^4}{2^4} \\ &=& \frac{1}{16} \end{array}\end{align*}

The Power of a Quotient Property says that 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*}\left( \frac{a}{b} \right)^n =\frac{a^n}{b^n}\end{align*}

Let’s look at an example.

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

First, apply the product of a quotient property.

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

Next, expand to simplify.

\begin{align*}\begin{array}{rcl} \frac{5^4}{3^4} &=& \frac{5 \cdot 5 \cdot 5 \cdot 5}{3 \cdot 3 \cdot 3 \cdot 3} \\ &=& \frac{625}{81} \end{array}\end{align*}

The answer is \begin{align*}\frac{625}{81}\end{align*}.

Let’s look at another example.

Simplify:\begin{align*}\left( \frac{3k}{2j} \right)^4\end{align*}.

First, apply the product of a quotient property.

\begin{align*}\left( \frac{3k}{2j} \right)^4 = \frac{(3k)^4}{(2j)^4}\end{align*}

Next, expand to simplify.

\begin{align*}\begin{array}{rcl} \frac{(3k)^4}{(2j)^4} &=& \frac{3^4 k^4}{2^2 j^4} \\ &=& \frac{3 \cdot 3 \cdot 3 \cdot 3 \cdot k \cdot k \cdot k \cdot k}{2 \cdot 2 \cdot 2 \cdot 2 \cdot j \cdot j \cdot j \cdot j} \\ &=& \frac{81k^4}{16j^4} \end{array}\end{align*}

The answer is \begin{align*}\frac{81k^4}{16j^4}\end{align*}.

Guided Practice

Simplify the following quotient.

\begin{align*}\left( \frac{-4x}{3y} \right)^3\end{align*}

First, apply the product of a quotient property.

\begin{align*}\left( \frac{-4x}{3y} \right)^3 = \frac{(-4x)^3}{(3y)^3}\end{align*}

Next, expand to simplify.

\begin{align*}\begin{array}{rcl} \frac{(-4x)^3}{(3y)^3} &=& \frac{-4 \cdot -4 \cdot -4 \cdot x \cdot x \cdot x}{3 \cdot 3 \cdot 3 \cdot y \cdot y \cdot y} \\ &=& \frac{-64x^3}{27y^3} \end{array}\end{align*}

The answer is \begin{align*} \frac{-64x^3}{27y^3}\end{align*}.

Examples

Example 1

Simplify the quotient:\begin{align*}\left( \frac{4}{5} \right)^3\end{align*}.

First, apply the product of a quotient property.

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

Next, expand to simplify.

\begin{align*}\begin{array}{rcl} \frac{4^3}{5^3} &=& \frac{4 \cdot 4 \cdot 4}{5 \cdot 5 \cdot 5} \\ &=& \frac{64}{125} \end{array} \end{align*}

The answer is \begin{align*}\frac{64}{125}\end{align*}.

Example 2

Simplify the quotient:\begin{align*}\left( \frac{2a}{3b} \right)^2\end{align*}.

First, apply the product of a quotient property.

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

Next, expand to simplify.

\begin{align*}\begin{array}{rcl} \frac{2 a^2}{3^2b^2} &=& \frac{2 \cdot 2 \cdot a \cdot a}{3 \cdot 3 \cdot b \cdot b} \\ &=& \frac{4a^2}{9b^2} \end{array} \end{align*}

The answer is \begin{align*}\frac{4a^2}{9b^2}\end{align*}.

Example 3

Simplify the quotient:\begin{align*}\left( \frac{a}{5b} \right)^3\end{align*}.

First, apply the product of a quotient property.

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

Next, expand to simplify.

\begin{align*}\begin{array}{rcl} \frac{a^3}{5^3b^3} &=& \frac{a \cdot a \cdot a}{5 \cdot 5 \cdot 5 \cdot b \cdot b \cdot b} \\ &=& \frac{a^3}{125b^3} \end{array}\end{align*}

The answer is \begin{align*}\frac{a^3}{125b^3}\end{align*}.

Follow Up

Credit: EvelynGiggles
Source: https://www.flickr.com/photos/evelynishere/3416643235/in/photolist-6cVbZZ-U9Lw-E1yU7-E1yTV-E1yTS-oFMxZU-eHJzg-zMGhb-oqufz3-opjQM7-opkap6-a6Kng2-8cdHx-KAD7r-FFjWR-opjEeV-a6KHYP-a6Nzgw-a6Nyny-a6NxdE-a6KEhk-a6Ntn9-a6KAkK-vhY6GT-v14gUh-63dvjH-oq3X4v-63hMym-vjBUFr-7vaZgc-63hJvy-63ibZj-a6KyTD-a6KymH-a6NpkY-a6Kw8B-a6Kv3Z-3e8aXt-a6NkNm-a6Kt5v-a6KrYT-a6KqCv-a6NgH1-a6Nfw9-a6Ndfu-a6KkbD-a6NuMJ-oGx2WT-oq3tYk-oEv8GN
License: CC BY-NC 3.0

Remember Professor Smith and the original 10 grams of cobalt?

Rakesh needs to figure out how much of the cobalt Professor Smith actually ended up using after she took one-third of one-third of one-third of one-third of 10 grams.

To figure out the number of grams in the sample, he must use monomials and powers.

First, set up the problem to model the information given in the story.

\begin{align*}10 \times \left( \frac{1}{3} \right)^4\end{align*}

Next, apply the product of a quotient property.

\begin{align*}10 \times \left( \frac{1}{3} \right)^4 = 10 \times \frac{1^4}{3^4}\end{align*}

Then, expand to simplify.

\begin{align*}\begin{array}{rcl} 10 \times \frac{1^4}{3^4} &=& 10 \times \frac{1 \cdot 1 \cdot 1 \cdot 1}{3 \cdot 3 \cdot 3 \cdot 3} \\ &=& 10 \times \frac{1}{81} \\ &=& \frac{10}{81} \end{array}\end{align*}

The answer is \begin{align*} \frac{10}{81}\end{align*}.

Professor Smith’s sample size was \begin{align*}\frac{10}{81}\end{align*} grams.

Video Review

https://www.youtube.com/watch?v=p_61XhXdlxI

Explore More

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^2j^8} \right)^5\end{align*}

Vocabulary

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.

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