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Supplemental Math: Study Aid1 chapters | 19 lessons

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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.

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 - 7
*m*^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.

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.

The only case when this would not be true is if *y* = 0. If the denominator of a fraction is 0, then the fraction is undefined. Therefore, substituting 0 for *y* in the example we have been using would give us 0^3 / 0^3. This would not equal 1 but would be undefined because of the 0 in the denominator.

Here are a few more quick examples so you can see the zero exponent rule in practice:

*t*^0 = 1- (
*x*^2 **y*^2) / (*x*^2 **y*^2) =*x*^(2 - 2) **y*^(2 - 2) =*x*^0 **y*^0 = 1 - 6
*m*^0*n*^2 = 6*n*^2

There are many rules that are used to simplify expressions in mathematics. The **zero exponent rule** is used to simplify terms with zero exponents. The rule states that any term with zero as an **exponent** is equal to one. The only time this is not true is if the **base** is zero. If the base is zero, the proof would require the zero to be in the denominator of a fraction. This result would be undefined. In summary, *x*^0 = 1, and *x* does not equal 0.

As this lesson concludes, you might have the skills necessary to:

- State and implement the zero exponent rule
- Solve an exponential equation that has zeros as exponents

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Supplemental Math: Study Aid1 chapters | 19 lessons

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