If you've entered the world of exponents, you may already know that a positive exponent has a base – the number at the bottom – and a smaller number on top, which represents the exponent. So for the number y^{x}, y is the base and x is the exponent. The number value of x can be either positive or negative, which tells you whether the exponent represents multiplication or division.

#### TL;DR (Too Long; Didn't Read)

To calculate a negative exponent, write it as a fraction with 1 in the numerator and the positive exponent in the denominator. For example, x^{-15} would be 1/x^{15} and 5^{-3} would be 1/5^{3}.

## A Recap of Positive Exponents

Before addressing how negative exponents work, here's a quick review of positive exponents. A positive exponent is like a ticker that tells you how many times to multiply 1 by the base number (the full-size number at the bottom of the exponent). So for example, x^{2} is the same as 1 × x × x (1 multiplied by x twice); y^{5} is the same as 1 × y × y × y × y × y (1 multiplied by y 5 times); and a = b^{x} would represent 1 multiplied by b x times. You don't have to know what number x or any of the other variables represent; you just need to understand the concept. Once you're comfortable with the idea of positive exponents, you're just one small step away from calculating negative exponents.

## Negative Exponents as Division

If a positive exponent represents multiplying the base number by itself, negative exponents represent how many times you'd divide 1 by the base number. So while x^{2} represents 1 × x × x, x^{-2} represents 1 ÷ x ÷ x; y^{5} represents 1 × y × y × y × y × y, and y^{-5} equals 1 ÷ y ÷ y ÷ y ÷ y ÷ y; and so on. After you work a few of these, you'll start to realize that there's an obvious pattern here. For example, x^{-2} = 1/x^{2} and y^{-5} = 1/y^{5}. So if it makes your calculations easier, you can simply rewrite negative exponents as positive exponent in the denominator of a fraction, with 1 as the numerator.

## Working Negative Exponents in Your Calculator

Every calculator's functions are different. But as a general rule any scientific calculator should have an exponent key, usually marked with the carat or ^ symbol. You can use this key to enter any exponent, positive or negative, into the calculator; simply input the base number, hit the ^ key, then input the exponent. If the exponent is negative, use the calculator's negative key, which might look like a minus sign in parentheses: (−).

## Zero Exponents

As you continue your math journey, you'll no doubt encounter zero as an exponent. You can work zero exponents just as you'd calculate any other exponent; all you have to do is remember that a positive number in the exponent represents how many times you multiply 1 by the base number, and a negative number in the exponent represents how many times you divide 1 by the base number. So if you had 5^{0}, you'd multiply 1 by 5... zero times. In other words, you're left with 1 on its own. If your teacher tried to trick you by giving you the exponent 5^{-0}, you'd divide 1 by 5 exactly zero times, so you'd be left with 1 on its own again.