# 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:

2^{4} = 16

2^{3} = 8

2^{2} = 4

2^{1} = 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 0^{0} is undefined. If it's not too late, don't think about this too hard. It'll make your head hurt.

### Multiplication

Sample: 2^{5} × 2^{7}

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 2^{5} × 2^{7} = 2^{12}. 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,* a*^{b} × *a*^{c} = *a*^{(b+c)}. This makes sense so long as you remember what exponents are abbreviating.

### Negative Exponents

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, 3^{1} × 3^{-1} would be 3^{1+(-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

5^{7} × 5^{-7} = 5^{7+-7} = 5 = 1. So 5^{-7} is the same as 1/5^{7}. Are you loving this stuff as much as we think you are?

### Division

Sample: 2^{5} ÷ 2^{2}

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 2^{3}.

In general, *a ^{b} ÷ a^{c} = 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: 4^{2} ÷ 4^{4}

This translates to: . See what we did there on the end? Always look for ways that an expression can be further simplified.

Sample: 6^{3} ÷ 6^{7} 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, 4^{3} and 5^{2}. 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?

### Exponentiation

Sample: (2^{5})^{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.

(2^{5})^{3} = 2^{15}

So, in general, (*a*^{b})^{c} = *a*^{b × c}*.*

### Raising Products to a Power

Sample: (6 × 7)^{3} = (6 × 7)(6 × 7)(6 × 7) = 6^{3} × 7^{3}.

In general, if *a* and *b* are real numbers and *c* is a whole number, (*a* × *b*)^{c} = *a*^{c} × *b*^{c}.

### Raising Quotients to a Power

Sample:

If *a* and *b* are real numbers and *c* is a whole number, .