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Zero Exponent: Rule, Definition & Examples

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  • 0:01 What Is the Zero…
  • 1:28 How the Rule Works
  • 3:55 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe
The zero exponent rule states that any term with an exponent of zero is equal to one. This lesson will go into the rule in more detail, explaining how it works and giving some examples. There will also be a quiz to test your knowledge.

What Is the Zero Exponent Rule?

The zero exponent rule is one of the rules that will help you simplify exponents. Let's first define some terms as they relate to exponents. When you have a number or variable raised to a power, the number (or variable) is called the base, while the superscript number is called the exponent, or power.

Usually, you'll see this written with the base being a normal-size number (or letter, if you're working with a variable). The exponent will be in a slightly smaller font, raised a little up above and to the right of the base. However, in some formats, like this one, you'll see the base, a mark called a caret that looks like an inverted V, and then the exponent. So if you have a base of 2 and an exponent of 3, we'll write that out here as 2^3 = 8.

Now that you know the terms, let's go back to the zero exponent rule. The zero exponent rule basically says that any base with an exponent of zero is equal to one. For example:

  • x^0 = 1
  • 5^0 = 1
  • 3^0 * a^0 = 1
  • 7m^0 = 7 * 1 = 7. The 7 is its own term, and in this problem, it's being multiplied by the second term (m^0). That's why the entire expression is not equal to 1. The only portion that will be equal to 1 is the portion with the exponent of 0.

How the Rule Works

There is a solid mathematical reason for why this works. It's not just some arbitrary rule that mathematicians made up to keep algebra students confused. In order to explain the zero exponent rule, we need to back up a bit and talk about the rule for dividing exponents.

When you are dividing exponents, you subtract the exponents in the denominator from the exponents in the numerator. As with other operations, the base must be the same before you can combine exponents. For example, y^5 / y^3 = y^2 because 5 - 3 = 2.

How does this relate to the zero rule? Well, if you have a division problem that looks like this - y^3 / y^3 - and you use the division rule, you get y^0 because 3 - 3 = 0. We also know from simple mathematics that anything divided by itself is one:

  • 2 / 2 = 1
  • 5436 / 5436 = 1
  • x / x = 1
  • y^3 / y^3 = 1

So, because y^3 / y^3 = 1 (according to mathematics) and y^3 / y^3 = y^0 (according to the division rule), you can also say that y^0 = 1. In other words, 1 = y^3 / y^3 = y^0; therefore 1 = y^0.

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