Integer Exponents

Whole Numbers

Just like multiplication is repeated addition, exponents are a shorthand notation for repeated multiplication. For example 2⁵ (spoken 2 to the 5th power or 2 to the 5th for short) means 2 times 2 times 2 times 2 times 2. 2 times 2 is 4, times another 2 makes 8, times another 2 makes 16, and times another one makes 32, so 2⁵=32. Exponents of 1 and 0 are kind of special. Anything to the 1 power is itself and anything except 0 to the 0 is 1. (0 to the 0 is undefined) These two might seem it bit puzzling at first. The first thing to realize when something like this puzzles you, is that it doesn't really have to totally make sense because it is a definition. Multiplying something by itself 1 times or 0 times doesn't really make any sense, so there is no need to make sense out of it. The idea of exponents is simply extended to numbers for which repeated multiplication no longer makes sense in a way that makes for a nice pattern and makes the properties of exponents still work.

One way you can think about these special exponents is to make the starting number be 1 and think of exponents as representing how many times you are multiplying 1 by the number. So with 2⁵ you are multiplying 1 by 2 five times to get 32. Then with 2¹ you are multiplying 1 by 2 once to get 2 and with 2⁰ you are multiplying 1 by 2 zero times which still leaves you with 1.

Another way to look at it is to look at the pattern going backwards.

3⁴=3*3*3*3
3³=3*3*3
3²=3*3

At each stage as we go to one power lower we divide by the base 3, so to go down to the next lower number 1, it would seem that we should divide by 3 again to get

3¹=3*3/3=3

and then for the next number down 0, we could divide again to get

3⁰=3/3=1.

This is also one good reason not to raise 0 to the 0, because 0/0 isn't 1.

Negative Numbers

Like with 0 and 1, multiplying something by itself a negative number of times doesn't really make sense, but there is something that follows the same pattern and has the same properties as positive exponents that is useful to make negative exponents be, and that is to make the negative mean to take the reciprocal. What this sort of is, is making raising to a negative power be repeated division. If we continue we the backwards pattern that we used to see what 1 and 0 mean as powers once we get to 1, we can divide again to get

3^-1=1/3
3^-2=(1/3)/3=1/3²

and continuing like this in general we get that

3^-n=1/3ⁿ

Of course there is nothing special about 3, so to make a proper definition out of this we can replace it with a letter and say

a^-n=1/aⁿ.

Another way of seeing that this definition makes sense is to look at the division property of exponents. With positive exponents you can simplify something like 5⁷/5³ by realizing that the top is 7 fives multiplied together and the bottom is 3 fives multiplied together, so the three fives downstairs will cancel with three of the fives upstairs to leave 4 leftover, so in effect what happens is that the exponents subtract. And nothing special here about 5, 7, and 3, you could do this with any numbers. But what if the top and bottom were reversed and instead you had 5³/5⁷? Then if you didn't have negative number exponents, this rule wouldn't work, but you could still cancel out 3 fives. It would just leave 4 leftover in the bottom instead of in the top and you would get 1/5⁴. Without negative exponents you need to have two different rules, one for when the exponent upstairs is bigger and one for when the exponent downstairs is bigger. But by making the above definition we only need one rule, because when you subtract a bigger number from a smaller number you get a negative number, and it exactly works here if a negative exponent means the reciprocal of the positive exponent.

Another interesting thing about negative exponents defined this way is that powers of ten exactly correspond to the place values in our number notation system, with the negative powers as the place values to the right of the decimal point. The number

1234.56789 means

1x10³+2x10²+3x10¹+4x10⁰+5x10^-1+6x10^-2+7x10^-3+8x10^-4+9x10^-5

Now that I think I have convinced you that this definition makes some sense, here's something extra about it that can sometimes be helpful for computing with it.

You can think of the exponent as divided into two parts, the positive number and the minus sign. The positive number tells you how many times you multiply the number by itself and the minus sign says to take the reciprocal and you can do those operations in either order. The way the definition is written, you raise to the power and then take the reciprocal, which is usually the best way to do it if you have a whole number. So for example when you raise 2 to the -3 you write

2^-3=1/2³=1/8,

in effect multiplying 2 times 2 times 2 to get 8 and then taking the reciprocal to that. But if you want to raise a fraction to a negative power, particularly one with a 1 in the numerator, it is easier to do it the other way around. You could raise 1/2 to the 4 power by writing down the definition like this

(1/2)^-4=1/(1/2)⁴=1/(1/16)=16,

inverting and multiplying in the last step, but you could do it easier by first taking the reciprocal and then raising to the power.

The reciprocal of 1/2 is 2.
2⁴=16