Simplifying Expressions With Negative Exponents
Learning Outcomes
- Use the properties of exponents to simplify products and quotients that contain negative exponents and variables
All the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.
Summary of Exponent Properties
If [latex]a,b[/latex] are real numbers and [latex]m,n[/latex] are integers, then
[latex-display]\begin{array}{cccc}\mathbf{\text{Product Property}}\hfill & & & {a}^{m}\cdot {a}^{n}={a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & & {\left({a}^{m}\right)}^{n}={a}^{m\cdot n}\hfill \\ \mathbf{\text{Product to a Power Property}}\hfill & & & {\left(ab\right)}^{m}={a}^{m}{b}^{m}\hfill \\ \mathbf{\text{Quotient Property}}\hfill & & & \frac{{a}^{m}}{{a}^{n}}={a}^{m-n},a\ne 0\hfill \\ \mathbf{\text{Zero Exponent Property}}\hfill & & & {a}^{0}=1,a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill & & & {\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}},b\ne 0\hfill \\ \mathbf{\text{Definition of Negative Exponent}}\hfill & & & {a}^{-n}=\frac{1}{{a}^{n}}\hfill \end{array}[/latex-display]
example
Simplify:
1. [latex]{x}^{-4}\cdot {x}^{6}[/latex]
2. [latex]{y}^{-6}\cdot {y}^{4}[/latex]
3. [latex]{z}^{-5}\cdot {z}^{-3}[/latex]
Solution
| 1. |
|
|
[latex]{x}^{-4}\cdot {x}^{6}[/latex] |
| Use the Product Property, [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]. |
[latex]{x}^{-4+6}[/latex] |
| Simplify. |
[latex]{x}^{2}[/latex] |
| 2. |
|
|
[latex]{y}^{-6}\cdot {y}^{4}[/latex] |
| The bases are the same, so add the exponents. |
[latex]{y}^{-6+4}[/latex] |
| Simplify. |
[latex]{y}^{-2}[/latex] |
| Use the definition of a negative exponent, [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex]. |
[latex]\frac{1}{{y}^{2}}[/latex] |
| 3. |
|
|
[latex]{z}^{-5}\cdot {z}^{-3}[/latex] |
| The bases are the same, so add the exponents. |
[latex]{z}^{-5 - 3}[/latex] |
| Simplify. |
[latex]{z}^{-8}[/latex] |
| Use the definition of a negative exponent, [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex]. |
[latex]\frac{1}{{z}^{8}}[/latex] |
try it
[ohm_question]146301[/ohm_question]
In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property of Exponents.
example
Simplify: [latex]\left({m}^{4}{n}^{-3}\right)\left({m}^{-5}{n}^{-2}\right)[/latex].
Answer:
Solution
|
[latex]\left({m}^{4}{n}^{-3}\right)\left({m}^{-5}{n}^{-2}\right)[/latex] |
| Use the Commutative Property to get like bases together. |
[latex]{m}^{4}{m}^{-5}\cdot {n}^{-2}{n}^{-3}[/latex] |
| Add the exponents for each base. |
[latex]{m}^{-1}\cdot {n}^{-5}[/latex] |
| Take reciprocals and change the signs of the exponents. |
[latex]\frac{1}{{m}^{1}}\cdot \frac{1}{{n}^{5}}[/latex] |
| Simplify. |
[latex]\frac{1}{m{n}^{5}}[/latex] |
try it
[ohm_question]146303[/ohm_question]
If the monomials have numerical coefficients, we multiply the coefficients, just as we did in Integer Exponents and Scientific Notation.
example
Simplify: [latex]\left(2{x}^{-6}{y}^{8}\right)\left(-5{x}^{5}{y}^{-3}\right)[/latex].
Answer:
Solution
|
[latex]\left(2{x}^{-6}{y}^{8}\right)\left(-5{x}^{5}{y}^{-3}\right)[/latex] |
| Rewrite with the like bases together. |
[latex]2\left(-5\right)\cdot \left({x}^{-6}{x}^{5}\right)\cdot \left({y}^{8}{y}^{-3}\right)[/latex] |
| Simplify. |
[latex]-10\cdot {x}^{-1}\cdot {y}^{5}[/latex] |
| Use the definition of a negative exponent, [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex]. |
[latex]-10\cdot \frac{1}{{x}^{1}}\cdot {y}^{5}[/latex] |
| Simplify. |
[latex]\frac{-10{y}^{5}}{x}[/latex] |
try it
[ohm_question]146304[/ohm_question]
In the next two examples, we’ll use the Power Property and the Product to a Power Property.
example
Simplify: [latex]{\left({k}^{3}\right)}^{-2}[/latex].
Answer:
Solution
|
[latex]{\left({k}^{3}\right)}^{-2}[/latex] |
| Use the Product to a Power Property, [latex]{\left(ab\right)}^{m}={a}^{m}{b}^{m}[/latex]. |
[latex]{k}^{3\left(-2\right)}[/latex] |
| Simplify. |
[latex]{k}^{-6}[/latex] |
| Rewrite with a positive exponent. |
[latex]\frac{1}{{k}^{6}}[/latex] |
try it
[ohm_question]146306[/ohm_question]
example
Simplify: [latex]{\left(5{x}^{-3}\right)}^{2}[/latex].
Answer:
Solution
|
[latex]{\left(5{x}^{-3}\right)}^{2}[/latex] |
| Use the Product to a Power Property, [latex]{\left(ab\right)}^{m}={a}^{m}{b}^{m}[/latex]. |
[latex]{5}^{2}{\left({x}^{-3}\right)}^{2}[/latex] |
| Simplify [latex]{5}^{2}[/latex] and multiply the exponents of [latex]x[/latex] using the
Power Property, [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]. |
[latex]25{k}^{-6}[/latex] |
| Rewrite [latex]{x}^{-6}[/latex] by using the definition of a negative
exponent, [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex]. |
[latex]25\cdot \frac{1}{{x}^{6}}[/latex] |
| Simplify |
[latex]\frac{25}{{x}^{6}}[/latex] |
try it
[ohm_question]146307[/ohm_question]
In the following video we show another example of how to simplify a product that contains negative exponents.
https://youtu.be/J9A-JlTXnsQ
To simplify a fraction, we use the Quotient Property.
example
Simplify: [latex]\frac{{r}^{5}}{{r}^{-4}}[/latex].
Answer:
Solution
|
[latex]\frac{r^5}{r^{-4}}[/latex] |
| Use the Quotient Property, [latex]\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex] . |
[latex]{r}^{5-(\color{red}{-4})}[/latex] |
| Be careful to subtract [latex]5-(\color{red}{-4})[/latex] |
|
| Simplify. |
[latex]4^9[/latex] |
try it
[ohm_question]146308[/ohm_question]
In the next video we share more examples of simplifying a quotient with negative exponents.
https://youtu.be/J5MrZbpaAGcLicenses & Attributions
CC licensed content, Original
- Question ID 146301, 146303, 146304, 146306, 146307, 146308. Authored by: Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Simplify A Product of Expressions with Neg Exponents (2 Methods). Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
- Ex 2: Simplify Exponential Expressions With Negative Exponents - Basic. Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
CC licensed content, Specific attribution
