Exponent Quotient Rule
Simple overview of the exponent quotient rule.
Exponent Quotient Rule Main Idea
For any base $a\ne 0$, the exponent quotient rule is
\[\frac{a^m}{a^n}=a^{m-n}\]Watch this section: introduction at 0:00.
Where The Rule Comes From?
Consider the following
\[\frac{x^6}{x^3}\]Let’s follow our nose and see what happens if we write this out explicitly.
\[\begin{align} \frac{x^6}{x^3} &= \frac{x\cdot x\cdot x\cdot x\cdot x\cdot x}{x\cdot x\cdot x} &\text{Expanded form.}\\ &=\left(\frac{x}{x}\right)\left(\frac{x}{x}\right)\left(\frac{x}{x}\right)\left(\frac{x\cdot x\cdot x}{1}\right) &\text{Pair up as many matching bases as possible.}\\ &= \left(1\right)\left(1\right)\left(1\right)\left(\frac{x\cdot x\cdot x}{1}\right) &\text{Replace }\left(\frac{x}{x}\right) \text{ with } 1.\\ &=\left(\frac{x\cdot x\cdot x}{1}\right) &\text{Multiplying by }1 \text{ has no effect on the result.}\\ &= x^3 &\text{Combine expanded form.} \end{align}\]Notice we get a similar result when subtracting the denominator power from the numerator power:
\[\frac{x^6}{x^3}=x^{6-3}=x^3\]Consider the following
\[\frac{x^2}{x^7}\]Let’s follow our nose and see what happens if we write this out explicitly.
\[\begin{align} \frac{x^2}{x^7} &= \frac{x\cdot x}{x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x} &\text{Expanded form.}\\ &=\left(\frac{x}{x}\right)\left(\frac{x}{x}\right)\left(\frac{1}{x\cdot x\cdot x\cdot x\cdot x}\right) &\text{Pair up as many matching bases as possible.}\\ &= \left(1\right)\left(1\right)\left(\frac{1}{x\cdot x\cdot x\cdot x\cdot x}\right) &\text{Replace }\left(\frac{x}{x}\right) \text{ with } 1.\\ &=\left(\frac{1}{x\cdot x\cdot x\cdot x\cdot x}\right) &\text{Multiplying by }1 \text{ has no effect on the result.}\\ &= \frac{1}{x^5} &\text{Combine expanded form.}\\ &= x^{-5} & \text{Use negative exponent rule to move up to numerator.} \end{align}\]Notice we get a similar result when subtracting the denominator power from the numerator power:
\[\frac{x^2}{x^7}=x^{2-7}=x^{-5}=\frac{1}{x^5}\]So regardless of whether the power is higher in the numerator or denominator the rule
\[\frac{a^m}{a^n}=a^{m-n}\]saves us time from writing everything out, replacing sets as forms of 1, and recombining the result.
Watch this section: where the rule comes from at 0:27.
Reading The Relationship Between The Exponents
We can use the two written out examples to get broader rules for the result based on any $m$ and $n$ pair.
\[\frac{a^m}{a^n}=a^{m-n}.\]If $m>n$, then $m-n>0$ and so our result has a positive power.
If $m<n$, then $m-n<0$ and so our result has a negative power.
If $m=n$, then we can match our base up $m$ times and replace them with a product of all one’s. Therefore, our result is $1$. When doing the subtraction for this scenario
\[\frac{a^m}{a^n}=a^{m-n}=a^0=1.\]Watch this section: relationship between $m$ and $n$ at 2:30.
Example 3: Several Bases
The third example has several bases:
\[\frac{x^4y^2z^8j}{x^3y^5j^6}\]Handle each base separately.
For the $x$ terms:
\[\frac{x^4}{x^3}=x^{4-3}=x\]For the $y$ terms:
\[\frac{y^2}{y^5}=y^{2-5}=y^{-3}=\frac{1}{y^3}\]The $z^8$ stays in the numerator because there is no matching $z$ power in the denominator. The $j$ in the numerator means $j^1$, so
\[\frac{j}{j^6}=j^{1-6}=j^{-5}=\frac{1}{j^5}\]Putting those pieces together gives
\[\frac{x^4y^2z^8j}{x^3y^5j^6}=\frac{xz^8}{y^3j^5}\]Watch this section: example 3 at 3:12.
Timestamp Guide
More Practice
For the timestamped list of quotient-rule examples already on the site, use the Exponent Quotient Rule Example Archive.
You can also review the exponents topic page or compare this rule with the exponent product rule companion.
The video description points to JoeCMath’s exponent property playlist: watch the playlist on YouTube.