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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

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

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:

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Additional Activities

Additional Practice Simplifying Expressions with Exponents of Zero

We learned in the video lesson that anything to a power of 0 is equal to 1 (unless the base is also 0) Use this knowledge, along with other rules of exponents such as bm bn = b(m + n) , (bm)/(bn) = b(m - n) and (bm) n = bmn to simplify the following problems. Assume all variables are positive numbers.

Problems

1) (27x5 y8 z3)/(2x2 y9 z )0

2) (3x7 - 2y5 )0 + π0

3) (7x0 y5 )/(y2 x)0

Solutions

1) 27x5 y8 z3 )/(2x2 y9 z)0

We could use many different exponent rules to simplify the expression inside of the parentheses first, but, since the entire expression inside of the parentheses is raised to a power of 0, the entire expression simplifies to 1.

2) (3x7 - 2y5 )0 + π0

We know that (3x7 - 2y5 )0 = 1 and also π0 = 1 (remember that π is just a number) and so the entire expression simplifies to 1 + 1 = 2


3) 7x0 y5 )/(y2 x)0

The entire denominator is raised to a power of zero, so the denominator simplifies to 1. In the numerator, the x0 = 1, but notice that the 7 and the y5 do not have a power of zero, so the expression simplifies to (7x0 y5)/(y2 x)0 = (7(1)y5)/1 = 7y5

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