Properties of Exponents
A Little Bit About Zero
0 raised to any positive exponent is 0. This makes sense, because if you multiply one or more copies of 0 together, you'll just get 0. Turns out it's hard for 0 to become anything other than 0. Even if he really applies himself.
Any nonzero number raised to the 0 power is 1. Think about it this way:
24 = 16
23 = 8
22 = 4
21 = 2
As the exponent drops by 1, the answer is divided in half. If we drop the exponent by 1 once more and divide the answer in half again, we get 2 = 1. We can't believe how many times you just dropped that exponent. Can't you be more careful?
0 is a troublesome number. We want 0 raised to a power to be zero, but we also want any number raised to the 0 power to be 1. There's no way to win! This means that 00 is undefined. If it's not too late, don't think about this too hard. It'll make your head hurt.
Sample: 25 × 27
This means that you need to multiply 5 copies of 2 together, and then multiply that result by 7 copies of 2 multiplied together. That's a total of 12 copies of 2. So 25 × 27 = 212. Why so many copies of 2? What are you, passing them out at a meeting?
If we have the same base with two different exponents and we're multiplying these numbers, as in the above example, the exponents get added together. In symbols, if a, b and c are real numbers, ab × ac = a(b+c). This makes sense so long as you remember what exponents are abbreviating.
So far, we have only looked at exponents that are positive integers. Let's try to figure out what a number would be when raised to a negative exponent.
Suppose we want to understand what 3-1 means. Well, let's use what we know about multiplication of exponents. We know that, because we add exponents during multiplication, 31 × 3-1 would be 31+(-1) = 3 = 1. This tells us that 3-1 is the multiplicative inverse, or reciprocal, of 3! So . Did you follow that? If not, double back and read this paragraph again until it sinks in. It won't kill you.
Now what happens if we take bigger powers? 5-7, for example. In this case, we'll look at
57 × 5-7 = 57+-7 = 5 = 1. So 5-7 is the same as 1/57. Are you loving this stuff as much as we think you are?
Sample: 25 ÷ 22
This means , so we're just canceling out two of our 2's. Buh-bye, guys. You shall be missed.
By reducing, our fraction equals 23.
In general, ab ÷ ac = a(b - c), because we start out with b copies of a, divide out c copies, and are left with b - c copies.
Note that a is not equal to 0.
Notice that if b > c, you are left with a positive exponent. But if b < c, you have a negative exponent. Which shouldn't stress you out any, as you now know what to do with them.
Sample: 42 ÷ 44
This translates to: . See what we did there on the end? Always look for ways that an expression can be further simplified.
Sample: 63 ÷ 67 means 3 copies of 6 divided by 7 copies of 6:
Cancel out 3 copies of 6 from the top and bottom of the fraction to get .
Be careful: In order to use the properties above, the base of the exponents has to be the same. We can't combine, for example, 43 and 52. 3 copies of 4 and 2 copies of 5 multiply to give you 3 copies of 4 and 2 copies of 5, and that's unfortunately as nice as it gets with exponent notation. Which isn't very nice. Hear that, Santa?
Sample: (25)3 means (2 × 2 × 2 × 2 × 2)3. You can't just add the 5 and the 3 together in this instance, because what we're actually being asked to do is take 3 copies of (2 × 2 × 2 × 2 × 2), or 15 copies of 2 multiplied together. Looks a little like we're going to be multiplying exponents here. In fact, it looks a lot like that.
(25)3 = 215
So, in general, (ab)c = ab × c.
Raising Products to a Power
Sample: (6 × 7)3 = (6 × 7)(6 × 7)(6 × 7) = 63 × 73.
In general, if a and b are real numbers and c is a whole number, (a × b)c = ac × bc.
Raising Quotients to a Power
If a and b are real numbers and c is a whole number, .