Dividing Fractions with Exponents

Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

A fraction can be viewed as a division problem with the numerator divided by the denominator. In this lesson, we'll look at how we divide a fraction when the numerator and denominator have exponents in them.

Fractions and Division

What is a fraction? We can view a fraction as part of a whole. Imagine having 1/4 of an apple. Instead of a whole apple you have a fraction of it. You have 1 out of 4 pieces that make up that apple.

Another way to view a fraction is as a division problem. This becomes more obvious when looking at an improper fraction such as 4/2. In this case you have four halves of an apple, which is the same as having two whole apples. In other words, 4 divided by 2 equals 2. This also works for proper fractions. Dividing proper fractions gives us a decimal. If we have 1/2, that's 1 divided by 2 which equals 0.5.

Now, let's complicate things a little further. What happens when the numerator, the top part of the fraction, and the denominator, the bottom part of the fraction, have exponents, meaning they are both raised to a power? For this scenario, we have a fraction that looks like the following:

5^2 / 4^3

During the rest of this lesson, we're going to see how to deal with fractions like this by learning how to divide fractions with exponents.

Like Bases

Before we get into looking at a fraction with exponents like the previous example, there are a couple special scenarios where dividing fractions with exponents has a trick to make it easier for us. The first of these scenarios is when we have a fraction with like bases.

Whole numbers raised to a power have two components, the base and the exponent. In our example fraction 5^2 / 4^3 we have two bases, 5 in the numerator and 4 in the denominator. If we had like bases, those two numbers would be the same. Let's look at the following example:

7^5 / 7^2

When you have a fraction with like bases raised to one or more powers, it's the same as having that base raised to the difference of the two powers. (Remember, in math, we find the difference between numbers by subtracting). We'll use our example above to see how this works:

7^5 / 7^2 = 7^(5 - 2) = 7^3 = 343

For problems like these, the exponent in the denominator is always subtracted from the exponent in the numerator. Sometimes the smaller number will be in the numerator, and you will end up with a negative exponent.

4^2 / 4^4 = 4^(2 - 4) = 4^(-2)

If this happens, you need to remember that a number raised to a negative exponent is the same as the reciprocal of that number raised to a positive exponent.

4^(-2) = 1/(4^2) = 1/16 = 0.0625

Like Exponents

Our second scenario for making the division of fractions with exponents easier occurs when you have two different bases with like exponents. A fraction with like exponents in the numerator and denominator is the same as having that whole fraction raised to a single power.

5^2 / 6^2 = (5/6)^2

From here on you just divide the fraction normally, and then raise the answer you get to that power.

(5/6)^2 = (0.83)^2 = 0.69

Remember, when you have a decimal raised to a power it works the same way as a whole number raised to a power. It's equal to that decimal multiplied by itself as many times as the exponent indicates.

(0.83)^2 = 0.83 * 0.83 = 0.69

(0.83)^3 = 0.83 * 0.83 * 0.83 = 0.57

No Like Components

If you have neither like bases nor like exponents, then there is no shortcut to solving the problem. In this case, you have to solve for the numerator and denominator separately, and then divide those answers you get. Let's look back at our very first example:

5^2 / 4^3

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