Simplifying Complex Expressions I
Learning Outcomes
- Simplify complex expressions using a combination of exponent rules
- Simplify quotients that require a combination of the properties of exponents
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 as we will be using them here to simplify various exponential expressions.
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 & & & {\Large\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({\Large\frac{a}{b}}\right)}^{m}={\Large\frac{{a}^{m}}{{b}^{m}}},b\ne 0\hfill \\ \mathbf{\text{Definition of Negative Exponent}}\hfill & & & {a}^{-n}={\Large\frac{1}{{a}^{n}}}\hfill \end{array}[/latex-display]
Expressions with negative exponents
The following examples involve simplifying expressions with negative exponents.
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}={\Large\frac{1}{{a}^{n}}}[/latex]. |
[latex]{\Large\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}={\Large\frac{1}{{a}^{n}}}[/latex]. |
[latex]{\Large\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]{\Large\frac{1}{{m}^{1}}\cdot \frac{1}{{n}^{5}}}[/latex] |
| Simplify. |
[latex]{\Large\frac{1}{m{n}^{5}}}[/latex] |
try it
[ohm_question]146303[/ohm_question]
If we multipy two expressions with numerical coefficients, we multiply the coefficients together.
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}={\Large\frac{1}{{a}^{n}}}[/latex]. |
[latex]-10\cdot {\Large\frac{1}{{x}^{1}}}\cdot {y}^{5}[/latex] |
| Simplify. |
[latex]{\Large\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 to simplify expressions with negative exponents.
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]{\Large\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{x}^{-6}[/latex] |
| Rewrite [latex]{x}^{-6}[/latex] by using the definition of a negative
exponent, [latex]{a}^{-n}={\Large\frac{1}{{a}^{n}}}[/latex]. |
[latex]25\cdot {\Large\frac{1}{{x}^{6}}}[/latex] |
| Simplify |
[latex]{\Large\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
The following examples involve solving exponential expressions with quotients.
example
Simplify: [latex]{\Large\frac{{\left({x}^{2}\right)}^{3}}{{x}^{5}}}[/latex].
Solution
|
[latex]{\Large\frac{{\left({x}^{2}\right)}^{3}}{{x}^{5}}}[/latex] |
| Multiply the exponents in the numerator, using the
Power Property. |
[latex]{\Large\frac{{x}^{6}}{{x}^{5}}}[/latex] |
| Subtract the exponents. |
[latex]x[/latex] |
try it
[ohm_question]146230[/ohm_question]
example
Simplify: [latex]{\Large\frac{{m}^{8}}{{\left({m}^{2}\right)}^{4}}}[/latex]
Answer:
Solution
|
[latex]{\Large\frac{{m}^{8}}{{\left({m}^{2}\right)}^{4}}}[/latex] |
| Multiply the exponents in the numerator, using the
Power Property. |
[latex]{\Large\frac{{m}^{8}}{{m}^{8}}}[/latex] |
| Subtract the exponents. |
[latex]{m}^{0}=1[/latex] |
try it
[ohm_question]146231[/ohm_question]
example
Simplify: [latex]{\left({\Large\frac{{x}^{7}}{{x}^{3}}}\right)}^{2}[/latex]
Answer:
Solution
|
[latex]{\left(\frac{{x}^{7}}{{x}^{3}}\right)}^{2}[/latex] |
| Remember parentheses come before exponents, and the
bases are the same so we can simplify inside the
parentheses. Subtract the exponents. |
[latex]{\left({x}^{7 - 3}\right)}^{2}[/latex] |
| Simplify. |
[latex]{\left({x}^{4}\right)}^{2}[/latex] |
| Multiply the exponents. |
[latex]{x}^{8}[/latex] |
try it
[ohm_question]146233[/ohm_question]
example
Simplify: [latex]{\left({\Large\frac{{p}^{2}}{{q}^{5}}}\right)}^{3}[/latex]
Answer:
Solution
Here we cannot simplify inside the parentheses first, since the bases are not the same.
|
[latex]{\Large{\left(\frac{{p}^{2}}{{q}^{5}}\right)}}^{3}[/latex] |
| Raise the numerator and denominator to the third power using the Quotient to a Power Property, [latex]{\Large{\left(\frac{a}{b}\right)}}^{m}={\Large\frac{{a}^{m}}{{b}^{m}}}[/latex] |
[latex]{\Large\frac{(p^2)^{3}}{(q^5)^{3}}}[/latex] |
| Use the Power Property, [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]. |
[latex]{\Large\frac{{p}^{6}}{{q}^{15}}}[/latex] |
try it
[ohm_question]146234[/ohm_question]
example
Simplify: [latex]{\Large{\left(\frac{2{x}^{3}}{3y}\right)}}^{4}[/latex]
Answer:
Solution
|
[latex]{\Large{\left(\frac{2{x}^{3}}{3y}\right)}}^{4}[/latex] |
| Raise the numerator and denominator to the fourth
power using the Quotient to a Power Property. |
[latex]{\Large\frac{{\left(2{x}^{3}\right)}^{4}}{{\left(3y\right)}^{4}}}[/latex] |
| Raise each factor to the fourth power, using the Power
to a Power Property. |
[latex]{\Large\frac{{2}^{4}{\left({x}^{3}\right)}^{4}}{{3}^{4}{y}^{4}}}[/latex] |
| Use the Power Property and simplify. |
[latex]{\Large\frac{16{x}^{12}}{81{y}^{4}}}[/latex] |
try it
[ohm_question]146235[/ohm_question]
example
Simplify: [latex]{\Large\frac{{\left({y}^{2}\right)}^{3}{\left({y}^{2}\right)}^{4}}{{\left({y}^{5}\right)}^{4}}}[/latex]
Answer:
Solution
|
[latex]{\Large\frac{{\left({y}^{2}\right)}^{3}{\left({y}^{2}\right)}^{4}}{{\left({y}^{5}\right)}^{4}}}[/latex] |
| Use the Power Property. |
[latex]{\Large\frac{\left({y}^{6}\right)\left({y}^{8}\right)}{{y}^{20}}}[/latex] |
| Add the exponents in the numerator, using the Product Property. |
[latex]{\Large\frac{{y}^{14}}{{y}^{20}}}[/latex] |
| Use the Quotient Property. |
[latex]{\Large\frac{1}{{y}^{6}}}[/latex] |
try it
[ohm_question]146893[/ohm_question]
[ohm_question]146241[/ohm_question]
For more similar examples, watch the following video.
https://youtu.be/Mqx8AXl75UY
To conclude this section, we will simplify quotient expressions with a negative exponent.
example
Simplify: [latex]{\Large\frac{{r}^{5}}{{r}^{-4}}}[/latex].
Answer:
Solution
|
[latex]{\Large\frac{r^5}{r^{-4}}}[/latex] |
| Use the Quotient Property, [latex]{\Large\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]r^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/J5MrZbpaAGc
Contribute!
Did you have an idea for improving this content? We’d love your input.
Licenses & Attributions
CC licensed content, Original
- Question ID: 146230, 146231, 146233, 146234, 146235, 146893, 146241. Authored by: Lumen Learning. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
CC licensed content, Shared previously
- Ex 1: Simplify Expressions using Exponent Properties (Quotient / Power Properties). Provided by: ` Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
CC licensed content, Specific attribution
