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]\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]

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]

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example

Simplify: [latex]{\Large\frac{{m}^{8}}{{\left({m}^{2}\right)}^{4}}}[/latex]

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example

Simplify: [latex]{\left({\Large\frac{{x}^{7}}{{x}^{3}}}\right)}^{2}[/latex]

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example

Simplify: [latex]{\left({\Large\frac{{p}^{2}}{{q}^{5}}}\right)}^{3}[/latex]

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example

Simplify: [latex]{\Large{\left(\frac{2{x}^{3}}{3y}\right)}}^{4}[/latex]

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example

Simplify: [latex]{\Large\frac{{\left({y}^{2}\right)}^{3}{\left({y}^{2}\right)}^{4}}{{\left({y}^{5}\right)}^{4}}}[/latex]

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For more similar examples, watch the following video.

	[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]

	[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]

	[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]

	[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]

Appendix C: Further Techniques

Learning Outcomes

Summary of Exponent Properties

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