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Working With Negative Exponents

Using negative exponents is really quite simple; it just means to use the reciprocal of the base number. Generally, that just means flipping the fraction upside down (or making the whole number into a fraction.)

Here's what we mean:

becomes

becomes

What you can see there is the negative sign flips the base number, but the exponent remains the same! (Remember, when the base "flips," a whole number becomes a number over 1 since a whole number can also be written as the numerator over the denominator of 1, such as: 14/1 is the same as 14!)

So, for any negative exponent, if the base number is a whole number, make it a fraction. If it is a fraction, make it a whole number. And leave the exponent the same, just remove the negative sign!

Here is an example when the base is a fraction:

We still do the same steps: Flip the fraction (2/3 becomes 3/2), remove the negative sign from the exponent, and write our answer:

 

Also note that there will sometimes be fractions in the exponent! These will look like this:

If you read our previous tutorial, Fractions in Exponents, then you know that this is simply a root problem.

is the same as .
But there is a negative first, so we use the above rules.

First, we get rid of the negative in the exponent: .

Next, we do the rest of the exponent:

And that's it! (You can enter this into your calculator to get something close to 0.57735.) And you're done!

While there are several possibilities for problems with negative exponents, they all will follow along the same guidelines. You simply need to determine which steps need to be done to get the problem so that it looks more familiar to you, and then you will be able to quickly solve it!

Find another math lesson at the Math Lesson and Tutorial Center!


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