Fun With Exponents
- Jordan Neuman
- Aug 6
- 2 min read
Fun With Exponents
Jordan Neuman
20 minutes ago
2 min read
Why does 5^0 = 1? In fact, why is every non-zero number raised to the power of zero equal one? Let us look at some simple examples and figure out why this principle exists.
Five squared, written as 5^2, can be rewritten as 5 x 5. The solution to this equation is 25. 5^3, or five cubed, can be written as 5 x 5 x 5. This equals 125. We can see that every time we add one to the exponent our answer will multiply by five.
But how about when we go backward? Since division is the "opposite" of multiplication (technically an inverse operation), we see that each time we reduce the exponent by one, we are dividing by 5. So, just as 5^3 divided by 5^2 is 5 (125 ÷ 25= 5), 5^1 must be the number 5. So, 5^1 divided by 5^1 must equal 1 because any number divided by itself is equal to 1. Yet, to be consistent with what we have learned about exponents, 5^0 must equal 1 so that when we go up from 5^0 to 5^1, our answer is five times the previous answer. If 5^1 is equal to 5, 5^0 must be equal to 1. And this relationship must also hold for all non-zero numbers.
A consequence of this rule is the ease with which one can multiply or divide large numbers with the same base. In our example our base was 5. If you are multiplying, you merely add the bases. So, 5^3 multiplied by 5^2 would equal 5^5. 5^5 divided by 5^2 would equal 5^3. Again, you can see the effect of the zero power. 5^3 divided by 5^0 would equal 5^3, which is the equivalent of dividing by 1.
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