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Properties of Exponents

By Mandi Elam

The interaction between operations and exponents have certain properties that can help us when dealing with multiple exponents. These properties allow us to simplify equations and expressions.

Why is this concept useful?

We use the properties of exponents to simplify expressions and solve equations where more than one exponential term has the same base.

Where does this concept fit into the curriculum?

Grade 8

How We Use Properties of Exponents:

Here is a list and examples of some of the properties of exponents:

Product of Powers: am× an = am+nExample: 23 × 25 = 28Quotient of Powers: aman=am−nExample: 3934=39−4=35Power of a Power: (am)n=am×nExample: (43)2=43×2=46Power of a Product: (ab)m=ambmExample: (5×7)3=57×53Power of a Quotient: (ab)m=ambmExamples: (32)4=3423Zero Exponent: a0=1Example: 110=1Negative Exponent∗: a−m−1amExample: 6−2 = 162Rational Exponent: amn=amnExample: 523=523Reciprocal of a fraction: a−1=1aExample: 4−1=14 \begin{array}{l} {\rm Product\ of\ Powers:}\ a^{m} \times \ a^{n} \ =\ a^{m+n}\\ {\rm Example:}\ 2^{3} \ \times \ 2^{5} \ =\ 2^{8}\\ \\ {\rm Quotient\ of\ Powers:}\ \frac{a^{m}}{a^{n}} =a^{m-n}\\ {\rm Example:}\ \frac{3^{9}}{3^{4}} =3^{9-4} =3^{5}\\ \\ {\rm Power\ of\ a\ Power:}\ \left( a^{m}\right)^{n} =a^{m\times n}\\ {\rm Example:}\ \left( 4^{3}\right)^{2} =4^{3\times 2} =4^{6}\\ \\ {\rm Power\ of\ a\ Product:}\ ( ab)^{m} =a^{m} b^{m}\\ {\rm Example}:\ ( 5\times 7)^{3} =5^{7} \times 5^{3}\\ \\ {\rm Power\ of\ a\ Quotient:} \ \left(\frac{a}{b}\right)^{m} =\frac{a^{m}}{b^{m}}\\ {\rm Examples:} \ \left(\frac{3}{2}\right)^{4} =\frac{3^{4}}{2^{3}}\\ \\ {\rm Zero\ Exponent:}\ a^{0} =1\\ {\rm Example:}\ 11^{0} =1\\ \\ {\rm Negative\ Exponent^{*} :}\ a^{-m} -\frac{1}{a^{m}}\\ {\rm Example:}\ 6^{-2} \ =\ \frac{1}{6^{2}}\\ \\ {\rm Rational\ Exponent:} \ a^{\frac{m}{n}} =\sqrt[n]{a^{m}}\\ {\rm Example:}\ 5^{\frac{2}{3}} =\sqrt[3]{5^{2}}\\ \\ {\rm Reciprocal\ of\ a\ fraction:} \ a^{-1} =\frac{1}{a}\\ {\rm Example:}\ 4^{-1} = \frac{1}{4}\\ \\ \end{array}

Sample Math Problems

1. Problem:

Find the value of m in the expression:

1117÷119=11m11^{17} \div 11^{9} = 11^m

Solution:

This problem uses the quotient of powers property:

aman=am−n\frac{a^{m}}{a^{n}} =a^{m-n}

1117119=1117−9=118\frac{11^{17}}{11^{9}} =11^{17-9} =11^{8}

This means that:

118=11b11^{8} =11^{b}
and so
b=8b = 8

2. Problem:

Evaluate and give the solution using only positive exponents:

(13−2)3\left( 13^{-2}\right)^{3}

Solution:

First, we simplify using the Power of a Power Property:

(13−2)3=13−6(13^{-2})^{3} =13^{-6}

Next, use the Negative Exponent Property to make the exponent a positive number:

13−6=113613^{-6} =\frac{1}{13^{6}}

3. Problem:

Rewrite

64m364m^{3}
as an exponential expression with a base of
4m4m

Solution:

Looking at

64m364m^{3}
and
4m4m
, it looks like we are using the Power of a Product rule:

(ab)m=ambm( ab)^{m} =a^{m} b^{m}

In order to be sure, we need to evaluate

43=644^{3} =64
. This means that
64m364m^{3}
can be written as
43m34^{3} m^{3}
, which is:
(4m)3(4m)^{3}
.

4. Problem:

Simplify

253225^{\frac{3}{2}}

Solution:

As the exponent is rational, this means we will be using the Rational Exponent Property:

amn=amna^{\frac{m}{n}} =\sqrt[n]{a^{m}}

8132=253281^{\frac{3}{2}} =\sqrt[2]{25^{3}}

=15,6252=\sqrt[2]{15,625}

=125=125

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Practice Math Problems

1.

Find the value of mi n the expression 55×52=5mFind\ the\ value\ of\ mi\ n\ the\ expression\ 5^{5} \times 5^{2} =5^{m}

2.

Evaluate and give the solution using only positive exponents (4−3)5Evaluate\ and\ give\ the\ solution\ using\ only\ positive\ exponents\ \left( 4^{-3}\right)^{5}

3.

Rewrite 32m5 as an exponential expression with a base of 2mRewrite\ 32m^{5} \ as\ an\ exponential\ expression\ with\ a\ base\ of\ 2m

4.

Simplify 826Simplify\ 8^{\frac{2}{6}}

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