Simplify Expressions With Negative Exponents

Simplify expressions with negative exponents. Here's an expression x squared divided by x to the fifth power. Okay. You might say the exponents have the same base, x. But there aren't any negative exponents here. Don't worry. We'll get to that. But first, let's simplify this expression. As we already noticed, the bases are the same. And the rule is that if the bases are the same, we can just subtract the exponent in the denominator from the exponent in the numerator. To minus five is minus three. This now reads as x to the minus three. There's our negative exponent. But what does that mean? How do we interpret this expression? And how might we do an actual calculation? Let's rewind this and go back to the original expression. We'll keep the work we just did in mind. But let's try working this again, but in a different way. Let's expand the exponents into the base, multiplied by itself, the number of times indicated by the exponent. So x squared becomes x times x and x to the fifth power becomes x times x times x times x times x. Now, let's cancel out the common factors. The two in the numerator get matched up with two in the denominator, and we're left with one over x times x times x. And there's the answer, one over x to the third. So the first time we did this, we got x to the minus three, and now we just got one over x to the third. Which is it? I mean, this looks like two different answers. But are these really different, or are they two different ways of saying the same thing? X divided by x is one. So we can match up the x's and cancel them out because it equals to one. That's it. They are the same thing. X to the minus three is the same as one over x to the third. Anytime you see a base with a negative exponent, you can rewrite it as one over the base raised to the positive version of the exponent. Or as we like to say, x to the minus n equals one over x to the n. You'll remember that anytime you take a number and divide one by it. We'll call that the reciprocal. In this case, we are taking the base number with its exponent and making it into its reciprocal, only with the exponent itself with a reverse sign. Also, if you have a regular old positive exponent, you can take the reciprocal and make that exponent negative, or x to the n equals one over x to the minus n. And this is not just a fun fact. It's another tool in our math toolbox that we can use to simplify expressions and solve problems. Let's try it out, okay? Here's a tricky little expression. X squared times y divided by x times y to the third power. We've got a couple of xs and a couple of ys. One of the xs has an exponent, but the other x doesn't. And one of the ys has an exponent, but the other y doesn't. But we know that take any number and raise it to the power of one, and you still get the same number. It's like every number has an implied exponent of one. So let's show our implied exponents on the terms that seemingly don't have any, just to sort of even things out. The plain old garden variety y in the numerator is actually y to the first in disguise. In the unassuming x and the denominator is secretly x to the first. Everybody's got an exponent now. Now, we'll reorder the terms and do the division trick where we subtract the denominators from the numerators. But we can only do this if the bases are the same. So x is now raised to two minus one, and y is raised to one minus three. Do the math and we get x to the first and y to the negative two. Of course, x to the one is just x. So it's now x times y to the minus two. And here we are, face to face with another negative exponent. But this time we're ready. We can rewrite the y to the negative two as one over y squared. Change the x to x over one and multiply it by the one over y squared, and we get x over y squared. Our original expression boil down to similar terms, and absolutely no negative exponents.