Exponents & Roots: Definition & Examples

Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Exponents and roots are used to make numbers much larger or much smaller. In this lesson, we'll learn how to work with exponents and roots. We'll practice a variety of problems involving both, including examples involving negative numbers.

Defining Exponents and Roots

Sometimes a number is just a number. But sometimes it's carrying a small number on its shoulder like this: 53. That 3 is an exponent. With exponents, we're taking a number and multiplying it by itself. How many? Well, that is what the exponent tells us. In this case, it is saying we want to multiply three 5s together, or 5 * 5 * 5.


Exponent image


Exponents can get us big numbers. But what if we see something like this:


Radical image


That symbol over the 27 is called a radical symbol and it means we want to find a root of 27. The root of a number is the number that can be multiplied a certain amount of times to get us that number under the radical symbol. So roots get us to the root of a number. That little three in the radical means that we want to find the number that when multiplied by itself three times equals 27.

If there's no number over the radical symbol, it's presumed to be a 2, which is called a square root. Let's look at examples of exponents and roots.

Solving Exponents

Remember that 53? We said that's 5 * 5 * 5. So 53 = 125.

Exponents make numbers explode very quickly. If we have 124, we want to multiply four 12s together, or 12 * 12 * 12 * 12. The first 12 * 12 = 144. 144 * 12 = 1728. 1728 * 12 = 20,736. So 124 = 20,736.

What happens if the base number is negative, as in -32? We do the same thing, but we need to include the negative sign. So this one is -3 * -3, which is positive 9.

If we had -33, it would be -3 * -3 * -3. A negative times a negative is positive. But then a positive times a negative is negative. So -33 = -27.

So far, we've only looked at expressions with positive exponents. What if the exponent is negative? Here's one: 4-2. What is this asking us to do? If a positive exponent means multiplying, then a negative exponent means the opposite of multiplying, which is dividing.

Really, we just can't have a negative exponent. To make it positive, we put the same expression (without the negative sign) as the denominator of a fraction:


1 over 4 squared


That will be 1/16.

There's one other exponent situation: 20. How can we multiply zero 2s together? We can't. But any time you see a 0 as an exponent, know that the answer is 1. So 20 = 1. Why is this so? It's called the zero exponent rule. It's a big complicated to explain, so just keep in mind that expressions with a 0 for an exponent are 1.

Solving Roots

Now that we've covered exponents, let's talk about roots. Let's start simple:


root 4


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