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Study Guides > Prealgebra

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 a,ba,b are real numbers and m,nm,n are whole numbers, then Product Propertyam⋅an=am+nPower Property(am)n=am⋅nProduct to a Power Property(ab)m=ambmQuotient Propertyaman=am−n,a≠0,m>naman=1an−m,a≠0,n>mZero Exponent Definitiona0=1,a≠0Quotient to a Power Property(ab)m=ambm,b≠0\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}
 

example

Simplify: (x2)3x5\frac{{\left({x}^{2}\right)}^{3}}{{x}^{5}}. Solution
(x2)3x5\frac{{\left({x}^{2}\right)}^{3}}{{x}^{5}}
Multiply the exponents in the numerator, using the Power Property. x6x5\frac{{x}^{6}}{{x}^{5}}
Subtract the exponents. xx
 

try it

[ohm_question]146230[/ohm_question]
   

example

Simplify: m8(m2)4\frac{{m}^{8}}{{\left({m}^{2}\right)}^{4}}.

Answer: Solution

m8(m2)4\frac{{m}^{8}}{{\left({m}^{2}\right)}^{4}}
Multiply the exponents in the numerator, using the Power Property. m8m8\frac{{m}^{8}}{{m}^{8}}
Subtract the exponents. m0{m}^{0}

 

try it

[ohm_question]146231[/ohm_question]
   

example

Simplify: (x7x3)2{\left(\frac{{x}^{7}}{{x}^{3}}\right)}^{2}.

Answer: Solution

(x7x3)2{\left(\frac{{x}^{7}}{{x}^{3}}\right)}^{2}
Remember parentheses come before exponents, and the bases are the same so we can simplify inside the parentheses. Subtract the exponents. (x7−3)2{\left({x}^{7 - 3}\right)}^{2}
Simplify. (x4)2{\left({x}^{4}\right)}^{2}
Multiply the exponents. x8{x}^{8}

 

try it

[ohm_question]146233[/ohm_question]
 

example

Simplify: (p2q5)3{\left(\frac{{p}^{2}}{{q}^{5}}\right)}^{3}.

Answer: Solution Here we cannot simplify inside the parentheses first, since the bases are not the same.

(p2q5)3{\left(\frac{{p}^{2}}{{q}^{5}}\right)}^{3}
Raise the numerator and denominator to the third power using the Quotient to a Power Property, (ab)m=ambm{\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}} (p2)3(q5)3\frac{{\left({p}^{2}\right)}^{3}}{{\left({q}^{5}\right)}^{3}}
Use the Power Property, (am)n=am⋅n{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}. p6q15\frac{{p}^{6}}{{q}^{15}}

 

try it

[ohm_question]146234[/ohm_question]
   

example

Simplify: (2x33y)4{\left(\frac{2{x}^{3}}{3y}\right)}^{4}.

Answer: Solution

(2x33y)4{\left(\frac{2{x}^{3}}{3y}\right)}^{4}
Raise the numerator and denominator to the fourth power using the Quotient to a Power Property. (2x3)4(3y)4\frac{{\left(2{x}^{3}\right)}^{4}}{{\left(3y\right)}^{4}}
Raise each factor to the fourth power, using the Power to a Power Property. 24(x3)434y4\frac{{2}^{4}{\left({x}^{3}\right)}^{4}}{{3}^{4}{y}^{4}}
Use the Power Property and simplify. 16x281y4\frac{16{x}^{2}}{81{y}^{4}}

 

try it

[ohm_question]146235[/ohm_question]
   

example

Simplify: (y2)3(y2)4(y5)4\frac{{\left({y}^{2}\right)}^{3}{\left({y}^{2}\right)}^{4}}{{\left({y}^{5}\right)}^{4}}.

Answer: Solution

(y2)3(y2)4(y5)4\frac{{\left({y}^{2}\right)}^{3}{\left({y}^{2}\right)}^{4}}{{\left({y}^{5}\right)}^{4}}
Use the Power Property. (y6)(y8)y20\frac{\left({y}^{6}\right)\left({y}^{8}\right)}{{y}^{20}}
Add the exponents in the numerator, using the Product Property. y14y20\frac{{y}^{14}}{{y}^{20}}
Use the Quotient Property. 1y6\frac{1}{{y}^{6}}

 

try it

[ohm_question]146893[/ohm_question] [ohm_question]146241[/ohm_question]
For more similar examples, watch the following video. https://youtu.be/Mqx8AXl75UY

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