The formalized definition of the quotient rule for exponents is…
\[\frac{x^{a}}{x^{b}} = x^{a-b}\]
How to Use the Quotient Rule for Exponents
- Make sure that the bases of the powers are the same.
- Make sure that the powers are divided by each other.
- Identify the exponents and subtract the denominator exponent from the numerator exponent.
- Write the simplified power with the result of Step 3 as the exponent of the original base.
You can NOT use the quotient rule for exponents if the bases are different. If the bases are different but the exponents are the same, you can use the reverse distributive property for exponents.
You can NOT use the quotient rule for exponents if the powers are added or subtracted from each other. If the powers are multiplied together, you can use the product rule for exponents.
Examples
\(\frac{2^{7}}{2^{3}} = 2^{4}\)
\(\frac{x^{5}}{x^{2}} = x^{3}\)
\(\frac{a^{2}}{a^{9}} = a^{-7}\)
Why It Works
To understand why the quotient rule for exponents works, let’s simplify \(\frac{a^{8}}{a^{3}}\) with order of operations instead of the quotient rule.
\(\frac{a^{8}}{a^{3}}\)
I’ll start by simplifying the powers. The exponent (8) tells me that I need to multiply the a in the numerator by itself 8 times. The exponent (3) tells me that I need to multiply the a in the denominator by itself 3 times.
\(\frac{aaaaaaaa}{aaa}\)
When I reduce the fraction, 3 of the a’s cancel out.
There are 5 a’s remaining in the numerator. I can rewrite that as \(a^{5}\).
So, that is why…
\(\frac{a^{8}}{a^{3}}=a^{5}\)
The quotient rule for exponents also works if you have numbers instead of variables.
\(\frac{3^{7}}{3^{5}}\)
Following the order of operations, I’ll simplify the stuff inside parentheses first. The exponent (7) tells me that I need to multiply the 3 in the numerator by itself 7 times. The exponent (5) tells me that I need to multiply the 3 in the denominator by itself 5 times.
\(\frac{3\times3\times3\times3\times3\times3\times3}{3\times3\times3\times3\times3}\)
When I reduce the fraction, 5 of the threes cancel out.
There are 2 threes remaining. I can rewrite that as \(3^{2}\).
So, that is why…
\(\frac{3^{7}}{3^{5}}=3^{2}\)
Reverse Quotient Rule for Exponents
We don’t use the the reverse quotient rule for exponents very often, but if you ever need to write a single power as a quotient of two powers, it would be helpful.
\[x^{a-b}=\frac{x^{a}}{x^{b}}\]
Why do the bases have to be the same?
The quotient rule for exponents is the exponent version of subtracting like terms. When you’re subtracting like terms, you can’t simplify \(3x-4y\) because the terms don’t have the same factors.
Similarly, when you’re dividing powers, you can’t simplify \(\frac{x^{3}}{y^{4}}\) because they don’t have the same base. This is really easy to see when you use order of operations to expand the powers.
\(\frac{x^{3}}{y^{4}}\)
The exponent (3) tells me that I need to multiply x by itself 3 times. The exponent (4) tells me that I need to multiply y by itself 4 times.
\(\frac{xxx}{yyyy}\)
There are 3 x’s and 4 y’s, which can be rewritten as \(\frac{x^{3}}{y^{4}}\).
However, this is the exact same expression that we were given. It can’t be simplified any further because the variables are not the same.
What if I have more than two powers?
If you have more than two powers, you can still subtract the exponents. You just have to make sure that ALL of the powers share the same base.
When you subtract the exponents, add all of the exponents in the numerator together first and then subtract all of the exponents in the denominator.
\(\frac{(7^{2})(7^{6})}{7^{3}}=7^{5}\)
\(\frac{x^{5}}{(x^{8})(x^{4})}=x^{-7}\)
If you simplify an example using order of operations, you can see WHY the exponents can be subtracted even if you have more than two powers…
\(\frac{x^{5}}{(x^{8})(x^{4})}\)
I’ll start by simplifying each of the powers. The exponent (5) tells me to multiply the first x by itself five times. The exponent (8) tells me to multiply the second x by itself eight times. The exponent (4) tells me to multiply the third x by itself four times.
\(\frac{xxxxx}{(xxxxxxxx)(xxxx)}\)
When I reduce the fraction, 5 of the x’s cancel out.
\(\frac{xxxxx}{(xxxxxxxx)(xxxx)}=\frac{1}{xxxxxxx}\)
There are 7 x’s remaining in the denominator. I can rewrite that as \(x^{-7}\).
So, that is why…
\(\frac{x^{5}}{(x^{8})(x^{4})}=x^{-7}\)
What if one of the bases doesn't have an exponent?
Remember that any number written without an exponent technically has an “invisible” exponent of 1. So, assuming that the bases are the same, you can just subtract 1 from the other exponent (or vice versa).
\(\frac{a^{3}}{a}=a^{2}\)
\(\frac{x}{x^{6}}=x^{-5}\)
\(\frac{y}{y}=y^{0}\)
This example shows WHY we can add an invisible exponent of 1 to the other exponent…
\(\frac{z^8}{z}\)
The exponent (8) tells me that I need to multiply the first z by itself 8 times.
\(\frac{z^8}{z}=\frac{zzzzzzzz}{z}\)
When I reduce the fraction, 1 of the z’s cancels out.
There are 7 z’s remaining in the numerator. I can rewrite that as \(z^{7}\).
So, that is why…
\(\frac{z^8}{z}=z^7\)
What if there's a negative exponent?
If one (or both) of the powers has a negative exponent, you can still subtract the exponents.
You just need to make sure to follow the rules for subtracting negative numbers and remember what a negative exponent means.
\(\frac{6^{3}}{6^{-5}}=6^{8}\)
\(\frac{x^{-3}}{x^{-7}}=x^{4}\)
\(\frac{y^{-1}}{y^{4}}=y^{-5}\)
This example shows WHY we can subtract negative exponents…
\(\frac{y^{-1}}{y^{4}}\)
The exponent (-1) tells me that I need to multiply y by itself once and move it to the denominator. The exponent (4) tells me that I need to multiply y by itself 4 times.
\(\frac{1}{(y)(yyyy)}\)
The fraction can’t be reduced and there are 5 y’s multiplied together in the denominator, so…
\(\frac{y^{-1}}{y^{4}}=y^{-5}\)
If it’s easier for you, you could also use the negative exponent rule to make the denominator exponent negative and then use the product rule for exponents.
What if there's a fractional exponent?
If there’s a fractional exponent, you can still subtract the exponents. You just need to make sure you follow the rules for subtracting fractions (or subtracting fractions and whole numbers).
\(\frac{x^{2}}{x^\frac{3}{2}}= x^\frac{1}{2}\)
\(\frac{x^\frac{1}{2}}{x^\frac{1}{4}}= x^\frac{1}{4}\)
To understand WHY this works, you first need to understand how to divide square roots and the relationship between roots and fractional exponents.
After you’ve reviewed those pages, come back to this example 🙂
\(\frac{x^{2}}{x^\frac{3}{2}}\)
I’ll start by writing the exponents with common denominators. The common denominator is 2 and I don’t have to change the first exponent because it already has a denominator of 2.
The exponent 2 can be rewritten as \(\frac{4}{2}\)
\(\frac{x^{2}}{x^\frac{3}{2}}=\frac{x^\frac{4}{2}}{x^\frac{3}{2}}\)
Next, I’ll simplify the powers. The exponent (\(\frac{4}{2}\)) tells me that I need to take the square root of \(x^4\). The exponent (\(\frac{3}{2}\)) tells me that I need to take the square root of \(x^3\).
\(\frac{x^\frac{4}{2}}{x^\frac{3}{2}}=\frac{\sqrt{xxxx}}{\sqrt{xxx}}\)
When I put all the x’s under the same square root symbol, I can reduce the fraction.
\(\frac{\sqrt{xxxx}}{\sqrt{xxx}}=\sqrt{\frac{xxxx}{xxx}}\)
Three of the x’s cancel out, leaving me with one x under the square root sign.
\(\sqrt{\frac{xxxx}{xxx}}=\sqrt{x}\)
That can be rewritten as \(x^\frac{1}{2}\), so…
\(\frac{x^{2}}{x^\frac{3}{2}}= x^\frac{1}{2}\)
