I've posted this in the past but have had a lot of interest in this, so will repost for those that missed it the first time.

Over the course of mathematics, we learn how to raise a quantity to an exponent. We know, for example, that xⁿ means we multiply x by itself n times. But suppose n is zero. How do we multiply a number by itself zero times? We are taught that any number raised to the zero power equals one, but rarely does anyone explain why this is the case.

One way to explain the zero exponent phenomenon is to use rules for exponents. When multiplying like bases, exponents are added. For example, 3⁵ x 3⁰ = 3^{(5 + 0)}. Therefore 3⁵ x 3⁰ = 3⁵, so 3⁰ must equal 1.
When dividing like bases, exponents are subtracted. For example, (4⁶)/(4⁰) = 4^{(6 - 0)}. Therefore (4⁶)/(4⁰) = 4⁶, so 4⁰ must equal 1.
Another way to show that any base raised to the zero power is one is to examine some patterns. Notice the pattern in the following:

• 2⁴ = 16
• 2³ = 8
• 2² = 4
• 2¹ = 2
• 2⁰ = ?

Notice that as exponents decrease by 1, the result is divided by the base number 2. Continuing this pattern, 2⁰ = 1. This will hold true no matter what the base number is.
Finally, I can use the concept of limits to show that any number raised to the zero power is one. Suppose we take 5ⁿ, and we start with n = 1. We know that 5¹ = 5. Take the square root of 5, which is equivalent of 5^(1/2), which is approximately 2.24. Now take a smaller value for n, such as 1/3, which gives us 5^(1/3) = 1.71. Continue to take values for n smaller and smaller but not less than or equal to zero. You'll start to notice what is happening, 5^(1/10) = 1.17 , 5^(1/1000) = 1.002, 5^(1/100000) = 1.00002. Notice how the result is getting closer and closer to 1. We say that the limit as n approaches 0 of 5ⁿ = 1.
I just explained three methods that clarify why any number raised to the zero power equals one. Next time someone is puzzled by this fact, you can explain the reason behind this somewhat vague and often unexplained topic in mathematics.