How Can An Exponent Be Negative?

Review of Exponents

You already know that an exponent represents the number of times you have to multiply a number by itself. For example, \(2^4\) means \(2*2*2*2\). But what if your variable is being raised to a negative exponent? If you were given \(2^{-4}\), how would you multiply two by itself negative four times?

A negative exponent is equivalent to the inverse of the same number with a positive exponent. In other words:

$$ x^{-7}=\frac{1}{x^7} $$

There is nothing special about solving a problem that includes negative exponentials. It may only require an intermediate step to complete to make things simpler. The best way to get comfortable with the concept of negative exponents is to work a few example problems that use them. Here are some samples:

Samples:

$$ 3^{-2}=\frac{1}{3^2}=\frac{1}{9} $$ $$ 4^{-1}=\frac{1}{4^1}=\frac{1}{4} $$ $$ 2*4^{-2}=\frac{2}{4^2}=\frac{1}{8} $$ $$ \frac{1}{2^{-2}}=2^2=4 $$ $$ \frac{1}{5^{-1}}=5^1=5 $$

Perhaps you'd like to try solving some equations with negative exponents on variables. This works just like solving any other equation. Don't be scared by the fact that the exponent is negative. Just work it out and remember to "flip over" (invert) the number if it has a negative exponent.

Example:

Solve the following for x:

$$ \frac{1}{x^{-4}}=16 $$

Solution:

You can see a negative exponent, so what's the first thing you might want to do? I would try to get the exponent to be positive. So, following our definition, just flip over the factor with the negative exponent and make the exponent positive!

$$ \frac{1}{x^{-4}}=16 $$ $$ 1*x^4=16 $$ $$ x = \pm2 $$

Negative exponents are nothing to be afraid of. Remember that when you see a negative exponent you can put it on the other side of the fraction bar and make it a positive exponent. If you need more math help with this subject, you can see our math help message board and ask your question for free.