Negative Exponents

Pessimistic Exponents

You may or may not have noticed that you've yet to see any negative exponents. That's simply because negative exponents have a bit of a mind of their own. Don't worry. This section is here to save you. Get it? Pessimistic exponents. Negative exponents. Anyway…

If you don't know already, the main idea here is that exponents switch signs whenever they're moved to the opposite side of a fraction bar. No, that's not a bar where the fractions all hang out and have a good time. It's the line between the numerator and denominator.

Whenever a base is moved to the other side of a fraction bar, the exponent of that base switches from negative to positive. For instance, x ^-2 can be rewritten as . Similarly, is the same as x². Could life be any more awesome right now?

This brings us to a new rule: whenever like bases are divided, we subtract the exponent in the denominator from the one in the numerator.

This is called the quotient rule.

You might want to think of it this way: Multiplying out the numerator and denominator gives us .

That's five x's on top and just two x's below. Since x divided by x is 1, we can divide out two x's on the top and bottom. This leaves us with three x's, otherwise known as x³.

The same thing works in the other direction too—if the bigger exponent is in the denominator. This is important in case we get asked to simplify using only positive exponents. And you know that's gonna happen.

Sample Problem

Simplify using positive exponents: .

We told you so. We need to take the lovely exponent in the numerator and subtract it from the exponent in the denominator.

Now that wasn't so bad.

Sample Problem

Simplify: .

Yikes, we're going right for the jugular here; all three rules at once. First, we're going to take care of what's in the parentheses by adding exponents.

Next, we'll multiply 12 by 3 to get 36 before subtracting 4.

As long as we can add, multiply, and subtract, we're golden.

Anyone ready for the coefficients?

Sample Problem

Simplify using positive exponents: (-8z ^-6)(7z³) .

In case you weren't awake in the first section, coefficients are the numbers in front of or multiplied by the variables. In this problem, the -8 and 7 are coefficients. This tends to make things just a bit more confusing because we still need to treat the coefficients like normal numbers while applying exponent rules to the exponents.

In this particular problem, we multiply the -8 and 7 while adding the exponents.

Next.

Sample Problem

Simplify using positive exponents: .

If you went ahead and did all the work for this one before realizing it was all raised to the power of 0, we apologize. Okay, we don't really apologize. We may or may not be secretly laughing. But remember, anything raised to the 0 is 1. Anyway, here's our work for this problem-o.

We won't trick you with the next one. We actually do promise this time.

Sample Problem

Simplify using positive exponents: .

Whoa. Stepping it up is an understatement. This is crazy-looking, but it's definitely a good summary of all our rules up to this point. We're sure it's no problem for a well-trained Shmooper like you.

All we need to do is keep the x's with the x's and the y's with the y's, and deal with the coefficients separately. To start, we'll take care of the stuff inside the parentheses in both the numerator and denominator. We'll work inside out using the product and power rules.

We can also take care of those pesky coefficients by dividing 10 by 5.

Things are already looking up. Next, since we need positive exponents, we can use the quotient rule for the x's and y's separately. Of course, we're still inside the parentheses.

Finally, the -1 exponent can be multiplied to both of the other exponents as well as the whole number in the numerator. When you get really good, you'll see that a -1 exponent really just flips the fraction.

This can also be written as . Your call. It's correct either way.