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Simplifying Expressions Using the Quotient Property of Exponents

Simplifying Expressions Using the Quotient Property of Exponents

Earlier in this tutorial, we developed the properties of exponents for multiplication. We summarize these properties here. \(\require{cancel}\)

Definition: Summary of Exponent Properties for Multiplication

If \(a,\phantom{\rule{0.2em}{0ex}}b\) are real numbers and \(m,\phantom{\rule{0.2em}{0ex}}n\) are whole numbers, then

\(\begin{array}{cccc}\text{Product Property}\hfill & & & \hfill {a}^{m}\cdot {a}^{n}={a}^{m+n}\hfill \\ \text{Power Property}\hfill & & & \hfill \phantom{\rule{0.4em}{0ex}}{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}\hfill \\ \text{Product to a Power}\hfill & & & \hfill \phantom{\rule{0.5em}{0ex}}{\left(ab\right)}^{m}={a}^{m}{b}^{m}\hfill \end{array}\)

Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. In Mathematics 104, you learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help us work with algebraic fractions—which are also quotients.

Definition: Equivalent Fractions Property

If \(a,\phantom{\rule{0.2em}{0ex}}b,\phantom{\rule{0.2em}{0ex}}c\) are whole numbers where \(b\ne 0,\phantom{\rule{0.2em}{0ex}}c\ne 0,\) then

\(\cfrac{a}{b}=\cfrac{a\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}c}{b\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}c}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\cfrac{a\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}c}{b\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}c}=\cfrac{a}{b}\)

As before, we’ll try to discover a property by looking at some examples.

\(\begin{array}{cccccccccc}\text{Consider}\hfill & & & \hfill \frac{{x}^{5}}{{x}^{2}}\hfill & & & \text{and}\hfill & & & \hfill \frac{{x}^{2}}{{x}^{3}}\hfill \\ \text{What do they mean?}\hfill & & & \hfill \frac{x\cdot x\cdot x\cdot x\cdot x}{x\cdot x}\hfill & & & & & & \hfill \frac{x\cdot x}{x\cdot x\cdot x}\hfill \\ \text{Use the Equivalent Fractions Property.}\hfill & & & \hfill \frac{\cancel{x}\cdot \cancel{x}\cdot x\cdot x\cdot x}{\cancel{x}\cdot \cancel{x}\cdot 1}\hfill & & & & & & \hfill \frac{\cancel{x}\cdot \cancel{x}\cdot 1}{\cancel{x}\cdot \cancel{x}\cdot x}\hfill \\ \text{Simplify.}\hfill & & & \hfill {x}^{3}\hfill & & & & & & \hfill \frac{1}{x}\hfill \end{array}\)

Notice that in each case the bases were the same and we subtracted the exponents.

  • When the larger exponent was in the numerator, we were left with factors in the numerator and \(1\) in the denominator, which we simplified.
  • When the larger exponent was in the denominator, we were left with factors in the denominator, and \(1\) in the numerator, which could not be simplified.

We write:

\(\begin{array}{ccccc}\frac{{x}^{5}}{{x}^{2}}\hfill & \phantom{\rule{1em}{0ex}}& & & \hfill \frac{{x}^{2}}{{x}^{3}}\hfill \\ {x}^{5-2}\hfill & \phantom{\rule{1em}{0ex}}& & & \hfill \frac{1}{{x}^{3-2}}\hfill \\ {x}^{3}\hfill & \phantom{\rule{1em}{0ex}}& & & \hfill \frac{1}{x}\hfill \end{array}\)

Definition: Quotient Property of Exponents

If \(a\) is a real number, \(a\ne 0,\) and \(m,\phantom{\rule{0.2em}{0ex}}n\) are whole numbers, then

\(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\phantom{\rule{2em}{0ex}}m>n\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},\phantom{\rule{2em}{0ex}}n>m\)

A couple of examples with numbers may help to verify this property.

\(\begin{array}{cccc}\frac{{3}^{4}}{{3}^{2}}\stackrel{?}{=}{3}^{4-2}\hfill & & & \hfill \phantom{\rule{1.8em}{0ex}}\frac{{5}^{2}}{{5}^{3}}\stackrel{?}{=}\frac{1}{{5}^{3-2}}\hfill \\ \frac{81}{9}\stackrel{?}{=}{3}^{2}\hfill & & & \hfill \frac{25}{125}\stackrel{?}{=}\frac{1}{{5}^{1}}\hfill \\ \phantom{\rule{0.6em}{0ex}}9=9✓\hfill & & & \hfill \phantom{\rule{1.6em}{0ex}}\frac{1}{5}=\frac{1}{5}✓\hfill \end{array}\)

When we work with numbers and the exponent is less than or equal to \(3,\) we will apply the exponent. When the exponent is greater than \(3\), we leave the answer in exponential form.

Example

Simplify:

  1. \(\phantom{\rule{0.2em}{0ex}}\frac{{x}^{10}}{{x}^{8}}\)
  2. \(\phantom{\rule{0.2em}{0ex}}\frac{{2}^{9}}{{2}^{2}}\)

Solution

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

  
Since 10 > 8, there are more factors of \(x\) in the numerator.\(\frac{{x}^{10}}{{x}^{8}}\)
Use the quotient property with \(m>n,\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}\).Simplifying Expressions Using the Quotient Property of Exponents
Simplify.\({x}^{2}\)
  
Since 9 > 2, there are more factors of 2 in the numerator.\(\frac{{2}^{9}}{{2}^{2}}\)
Use the quotient property with \(m>n,\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}.\)Simplifying Expressions Using the Quotient Property of Exponents
Simplify.\({2}^{7}\)

Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.

Example

Simplify:

  1. \(\phantom{\rule{0.2em}{0ex}}\frac{{b}^{10}}{{b}^{15}}\)
  2. \(\phantom{\rule{0.2em}{0ex}}\frac{{3}^{3}}{{3}^{5}}\)

Solution

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

  
Since 15 > 10, there are more factors of \(b\) in the denominator.\(\frac{{b}^{10}}{{b}^{15}}\)
Use the quotient property with \(n>m,\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\)Simplifying Expressions Using the Quotient Property of Exponents
Simplify.\(\frac{1}{{b}^{5}}\)
  
Since 5 > 3, there are more factors of 3 in the denominator.\(\frac{{3}^{3}}{{3}^{5}}\)
Use the quotient property with \(n>m,\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\)Simplifying Expressions Using the Quotient Property of Exponents
Simplify.\(\frac{1}{{3}^{2}}\)
Apply the exponent.\(\frac{1}{9}\)

Notice that when the larger exponent is in the denominator, we are left with factors in the denominator and \(1\) in the numerator.

Example

Simplify:

  1. \(\phantom{\rule{0.2em}{0ex}}\frac{{a}^{5}}{{a}^{9}}\)
  2. \(\phantom{\rule{0.2em}{0ex}}\frac{{x}^{11}}{{x}^{7}}\)

Solution

  
Since 9 > 5, there are more \(a\)’s in the denominator and so we will end up with factors in the denominator.\(\frac{{a}^{5}}{{a}^{9}}\)
Use the Quotient Property for \(n>m,\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\)Simplifying Expressions Using the Quotient Property of Exponents
Simplify.\(\frac{1}{{a}^{4}}\)
  
Notice there are more factors of \(x\) in the numerator, since 11 > 7. So we will end up with factors in the numerator.\(\frac{{x}^{11}}{{x}^{7}}\)
Use the Quotient Property for \(m>n,\frac{{a}^{m}}{{a}^{n}}={a}^{n-m}.\)Simplifying Expressions Using the Quotient Property of Exponents
Simplify.\({x}^{4}\)

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