# What are the Laws of Exponents?

## Laws of Exponents

We’re going to look at seven different laws. The first is that any number raised to the first power is just itself. If you remember what an exponent is, it tells you how many times to multiply the base number times itself. If you’re multiplying the base number times itself just one time, then that’s just that number.

2 to the first power would just be 2, or -3 to the first power would just be -3. The second rule says that 1 raised to any power is 1, because all you’re doing is multiplying 1 times itself. No matter how many times you multiply 1 times itself, it will always be 1. For instance, like 1 to the third power.

That means 1 times itself three times. 1 times 1 is 1, times 1 is 1. It doesn’t matter how many times you multiply 1 times itself. You’ll always get 1. The third law says that any number raised to the zero power is 1. This is one of my personal favorites. I think it’s pretty cool. 8 raised to the zero power is just 1, or -10 raised to the zero power.

Again, just 1. Even say like 1/2 raised to the zero power. Still 1. Any number raised to the zero power is always 1. The fourth flaw is when you’re multiplying with the same bases. Notice A and A. Our bases are the same, but we have different exponents When you multiply numbers with the same bases, you add the exponents.

Let’s look at why. Say we are doing 2 cubed times 2 to the fourth. Well, what that really means is 2 cubed is 2 times itself three times (1, 2, 3 times) times 2 times itself four times (1, 2, 3, 4 times). We’ve multiplied 2 times itself a total of seven times.

Instead of going through all of this, we can just use that rule that when we multiply numbers with the same bases we simply add the exponent. Then you could simplify from there. The fifth rule is pretty similar to the fourth rule, except we’re dividing members at the same bases. Instead of adding our exponents, we subtract our exponents.

For example, if we were doing 2 to the fifth divided by 2 cubed, what we’re really doing is 2 times 2 times 2 times 2 times 2 divided by 2 times 2 times 2. That’s 2 to the fifth (2 times itself five times) divided by 2 times itself three times. We can simplify by cancelling. 2 divided by 2 is 1. 2 divided by 2 is 1. 2 divided by 2 is 1.

What we’re left with, then, is simply 2 squared (4). Instead of doing this, we can use our rule that when we’re dividing numbers with the same bases we simply subtract the exponents. That saves us a little bit of time. 2 to the fifth divided by 2 cubed. 5 minus 3 is 2, so it’s 2 squared, which is four.

It also works if your denominator, the exponent on your denominator, is larger. Let’s look at one of those. 2 squared divided by 2 cubed. You still subtract your exponents. It’s 2 squared minus 3, which is 2 to the -1. You would simplify that by bringing this to your denominator, one half.

The sixth rule says that when you raise a power to a power, like a to the n raised to the m, you multiply your exponents. For instance, with 3 squared cubed, what that really means is 3 squared times itself 3 times. That’s 3 squared times 3 squared times 3 squared. 3 squared times itself three times. 3 squared times 3 square times 3 squared is 3 to the sixth.

All you have to do is multiply your exponents. 2 times 3 is 6. 3 to the sixth. The seventh rule says that when you have operations inside parentheses that are raised to an exponent, each term inside those parentheses is raised so that exponent. For instance, 2 times 3 squared. That is 2 squared times 3 squared.

Again, you could simplify from there. Left off my n there. That’s important. Now, on this next one, same thing, just with division. A divided by B, that quantity raised to the n would be a to the n divided by b to the n.

Everything in the parentheses is raised to the n power. It would be like doing 4 divided by 3 cubed. 4 to the third divided by 3 to third. Again, you could simplify. Those are laws of exponents.