Combining Properties to Simplify Expressions
Learning Outcomes
- Simplify quotients that require a combination of the properties of exponents
We'll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.
Summary of Exponent Properties
If [latex]a,b[/latex] are real numbers and [latex]m,n[/latex] are whole numbers, 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,m>n\hfill \\ & & & \frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},a\ne 0,n>m\hfill \\ \mathbf{\text{Zero Exponent Definition}}\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 \end{array}[/latex-display]
example
Simplify: [latex]\frac{{\left({x}^{2}\right)}^{3}}{{x}^{5}}[/latex].
Solution
|
[latex]\frac{{\left({x}^{2}\right)}^{3}}{{x}^{5}}[/latex] |
| Multiply the exponents in the numerator, using the
Power Property. |
[latex]\frac{{x}^{6}}{{x}^{5}}[/latex] |
| Subtract the exponents. |
[latex]x[/latex] |
try it
[ohm_question]146230[/ohm_question]
example
Simplify: [latex]\frac{{m}^{8}}{{\left({m}^{2}\right)}^{4}}[/latex].
Answer:
Solution
|
[latex]\frac{{m}^{8}}{{\left({m}^{2}\right)}^{4}}[/latex] |
| Multiply the exponents in the numerator, using the
Power Property. |
[latex]\frac{{m}^{8}}{{m}^{8}}[/latex] |
| Subtract the exponents. |
[latex]{m}^{0}[/latex] |
try it
[ohm_question]146231[/ohm_question]
example
Simplify: [latex]{\left(\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(\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]{\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]{\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}}[/latex] |
[latex]\frac{{\left({p}^{2}\right)}^{3}}{{\left({q}^{5}\right)}^{3}}[/latex] |
| Use the Power Property, [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]. |
[latex]\frac{{p}^{6}}{{q}^{15}}[/latex] |
try it
[ohm_question]146234[/ohm_question]
example
Simplify: [latex]{\left(\frac{2{x}^{3}}{3y}\right)}^{4}[/latex].
Answer:
Solution
|
[latex]{\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]\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]\frac{{2}^{4}{\left({x}^{3}\right)}^{4}}{{3}^{4}{y}^{4}}[/latex] |
| Use the Power Property and simplify. |
[latex]\frac{16{x}^{2}}{81{y}^{4}}[/latex] |
try it
[ohm_question]146235[/ohm_question]
example
Simplify: [latex]\frac{{\left({y}^{2}\right)}^{3}{\left({y}^{2}\right)}^{4}}{{\left({y}^{5}\right)}^{4}}[/latex].
Answer:
Solution
|
[latex]\frac{{\left({y}^{2}\right)}^{3}{\left({y}^{2}\right)}^{4}}{{\left({y}^{5}\right)}^{4}}[/latex] |
| Use the Power Property. |
[latex]\frac{\left({y}^{6}\right)\left({y}^{8}\right)}{{y}^{20}}[/latex] |
| Add the exponents in the numerator, using the Product Property. |
[latex]\frac{{y}^{14}}{{y}^{20}}[/latex] |
| Use the Quotient Property. |
[latex]\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/Mqx8AXl75UYLicenses & Attributions
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
