Properties of Exponents

A Little Bit About Zero

If we raise 0 to any positive exponent, we still get 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 2⁰ = 1. We can't believe how many times you just dropped that exponent. Can't you be more careful?

So here's the deal:

2⁰ = 1
3⁰ = 1
15⁰ = 1
(-36.25)⁰ = 1

It's 1s all the way down: raise any number to the power of 0, and the answer is 1.

Well, except for one weird exception. What's 0⁰? Zero is a troublesome number. We want 0 raised to any power to be 0, 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.

Multiplication

What is 2⁵ × 2⁷?

This means that you need to multiply 5 copies of 2 together, and then multiply that result by 7 copies of 2. 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, then:

a^b × a^c = a^{(b + c)}

Negative Exponents

So far, we've 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. Let's use what we know about multiplying exponents. Since we add exponents during multiplication, 3¹ × 3^-1 would be 3^{1 + (-1)} = 3⁰ = 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? Like 5^-7, for example. In this case, we'll look at 5⁷ × 5^-7 = 5^{7 + (-7)} = 5⁰ = 1. So 5^-7 is the same as (¹/₅)⁷. Are you loving this stuff as much as we think you are?

Division

What's 2⁵ ÷ 2²?

This means , so we're just canceling out two of our 2s. Buh-bye, guys. You shall be missed.

After reducing, our fraction equals 2³.

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. 

Heads up, though: a can't be 0.

Notice that if b > c, you're 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 Problem

What's 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 Problem

What's 6³ ÷ 6⁷?

What this really means is "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. For example, we can't combine 4³ and 5². That's unfortunately as nice as it gets with exponent notation. Which isn't very nice. Hear that, Santa?

Exponentiation

What is (2⁵)³?

This really 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 = 2^{5 × 3} = 2¹⁵

So, in general, (a^b)^c = a^{b × c}.

Raising Products to a Power

What's (6 × 7)³?

Obviously we could just multiply 6 by 7 to get (42)³, but let's see what happens if we leave 'em separated.

(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 = a^c × b^c.

Raising Quotients to a Power

If a and b are real numbers and c is a whole number, . Just slap that exponent on the numerator and the denominator separately.