The Power of Zero: Simplifying Exponential Expressions

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  • 0:03 Zero is Different
  • 0:23 What Is an Exponent?
  • 1:06 Zero as an Exponent
  • 1:33 What's the Proof?
  • 4:25 Why Not Zero?
  • 4:50 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe
Raising a number to the power of zero is not the same as if the exponent is a number other than zero. This lesson will explain the rule involving using zero as an exponent and will give some examples of how it works.

Odd Man Out

The number zero is like one of those quirky people that you know that doesn't do things in quite the same way as everybody else. You know the type I'm talking about; they just march to the beat of a different drummer. Well, zero is the math world's version of that person. The number zero just doesn't work the same as all the other numbers. It follows its own set of rules.

What Is an Exponent?

Exponents are superscript numbers whose purpose is to state how many times a number should be multiplied to itself.

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Here you can see the number 2 with an exponent of three - or, most commonly, 2 to the third power. What it means is 2 multiplied to itself three times, or 2 * 2 * 2.

Exponents can also be written using a carat ^, which would look like 2^3 and would mean the same thing: 2 * 2 * 2.

Here's another example.

x^6 = x * x * x * x * x * x

Exponents work the same with both numbers and variables.

But Zero Is Different

However, as usual, zero is different. When used as an exponent, it does not work the same as other numbers.

The rule for zero as an exponent is that any number or variable (except zero itself) raised to the 0 power is equal to 1.

For example:

4^0 = 1

b^0 = 1

478^0 = 1

What's the Proof?

It's one thing for me to tell you that any number raised to the zero power is equal to one, but what if you didn't believe me? Well, here's the proof.

We need to start by looking at the rule for multiplying exponents.

n^3 * n^4 = (n*n*n) * (n*n*n*n) = n^7

n^6 * n^2 = (n*n*n*n*n*n) * (n*n) = n^8

Did you notice a shortcut in that? As long as the bases are the same, to multiply any two terms with exponents, just add the exponents. Since math is a logical study, most rules work the same in all cases. So, let's try with a zero exponent.

x^4 * x^0 = x^(4+0) = x^4

As you can see, nothing changed. So we need to ask ourselves the question, what is the only number that when you multiply something by it, nothing changes?

The only answer to that question is 1. When you multiply something by 1, it does not change. So the only thing to conclude with our example above is that x^0 = 1

x^4 * x^0 = x^4

x^4 * 1 = x^4

Therefore, x^0 = 1

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