Exponent Quotient Rule — Definition & Examples

(Video) Basic Exponent Properties

by PatrickJMT

Patrick JMT has a good video that explains the basic properties of exponents. Recall that an exponent is just a shorthand way of referring to repeated multiplication. $$5^3=5\cdot5\cdot5=125$$

$\large{\text{Summary}}$

Quotient Rule Overview

The quotient rule says that if we have exponents with the same base being divided, we can just subtract the exponents. Ultimately this is about dividing exponential expressions. The formula is below:

$$\boxed{\frac{a^m}{a^n}=a^{m-n}}$$

It has a few counterparts. The first is the product to a power rule. This rule says we can distribute a power between different exponents.

The second is the power rule. It says what to do if we raise a power to another power. Basically, we want to multiply the exponents.

Finally, there is the regular product rule. It says that if we multiply exponents with the same base, to add the exponents together.

Finally, let us note any of these rules work for positive exponents and negative ones. Solving many exponent problems comes down to one thing: apply the rules!

Now let's get into some quotient rule examples.

Examples

Suppose we wanted to simplify $\frac{3^4}{3^2}$. Our rule tells us: $$\frac{3^4}{3^2}=3^{4-2}=3^2$$ And to prove this we can write out the division. We cancel out some of the common factors of $3$. $$\begin{align} \frac{3^4}{3^2}&=\frac{3\cdot3\cdot3\cdot3}{3\cdot3}\\ &=\frac{\cancel{3}\cdot\cancel{3}\cdot3\cdot3}{\cancel{3}\cdot\cancel{3}}\\ &=3\cdot3\\ &=\boxed{3^2} \end{align}$$

Quotient Rule for Exponents

In order for the Quotient Rule for Exponents to work, the base in the numerator needs to be the same as the base in the denominator. You can't use this rule for a fraction like $\frac{13^2}{3^3}$ unless you somehow manipulate the fraction to make the numerator and denominator bases the same, since $13$ and $3$ are different numbers. Sometimes factoring the numerator or the denominator will make this possible.

Let's do one more example.

$$\textbf{A Second Example}$$

We will simplify $\frac{3^{15}}{3^{11}}+ \frac{2^{72}}{2^{73}}$.

First, let's simplify $\frac{3^{15}}{3^{11}}.$ If we use the variables from the graphic this means that $x=3, a=15,$ and $b=11.$ Now all we have to do is substitute these values into our Quotient Rule for Exponents formula: $$\begin{align} \frac{x^a}{x^b}&=x^{a-b}\\ \\ \frac{3^{15}}{3^{11}}&=3^{15-11}\\ &=3^{4}\\ &=81 \end{align}$$

So, $\frac{3^{15}}{3^{11}}=81.$ Now, let's simplify $\frac{2^{72}}{2^{73}}$. Here's a refresher on how to raise a number to a negative value.

$$\begin{align} \frac{x^a}{x^b}&=x^{a-b}\\ \\ \frac{2^{72}}{2^{73}}&=2^{72-73}\\ &=2^{-1}\\ &=\frac{1}{2} \end{align}$$

So, $\frac{2^{72}}{2^{73}}=\frac{1}{2}.$ Now let's simplify the entire expression. The first fraction is actually an improper fraction, and the second is a proper fraction.

$$\begin{align} \dfrac{3^{15}}{3^{11}}+ \dfrac{2^{72}}{2^{73}}&=81+\dfrac{1}{2} \\ &=81.5 \end{align}$$

Interestingly enough, the result is a mixed number. It can also be written as a decimal. Either of these give us a good sense of how big of numbers we are working with.

(Video) Quotient Rule for Exponents

by mathman1024

Here's a video by mathman1024 explaining the quotient rule for exponents.

$\large{\text{Summary}}$

The quotient rule for exponents states: $$\frac{x^a}{x^b}=x^{a-b}$$

To see why, let's look at the example $\frac{x^5}{x^2}$. If we expand these we see $$\frac{x^5}{x^2}=\frac{x\cdot x\cdot x\cdot x\cdot x}{x\cdot x}$$ We can cancel some of the $x$'s though. $$\begin{align} \frac{x^5}{x^2}&=\frac{x\cdot x\cdot x\cdot x\cdot x}{x\cdot x}\\ &=\frac{\color{#29A603}{\cancel{x}}\cdot \color{#29A603}{\cancel{x}}\cdot x\cdot x\cdot x}{\color{#29A603}{\cancel{x}}\cdot \color{#29A603}{\cancel{x}}}\\ &=x\cdot x\cdot x\\ &=x^3 \end{align}$$

So the quotient rule tells us that if we divide exponents with the same base then we just subtract the exponents. The base can be any real number except for $0$. We don't allow it to be $0$ because otherwise we may be dividing by zero.

Let's try an example: $\frac{x^{15}}{x^{7}}$. This problem breaks down if we apply the quotient rule.

$$\begin{align} \frac{x^{15}}{x^7}&=x^{15-7}\\ &=x^8 \end{align}$$

Try these next examples on your own first and then watch the video to see if you get the same answers.

1. $\frac{a^{100}}{a^{25}}$
2. $\frac{9^5}{9^3}$
3. $\frac{x^{11}y^{23}}{x^8y^{14}}$