This week in math I learned how to deal with numbers that have a negative exponent, when I was first tasked on solving a power with a negative exponent, I first thought that it’d be similar to positive exponents just with a negative, so did everyone else I was working with, but turns out they work completely differently.

When solving positive exponents, you take the base and multiply it by itself however many times the base says so, for example, is solved by doing 5 x 5 x 5 which gets you 125. On the other hand, would not get you -125 but instead would give you the answer or simplified.

The way you solve negative exponents is by flipping a sometimes hidden fraction, also known as getting the reciprocal, numbers not apart of any visible fractions always have a fraction under them which is always 1, this is typically never shown as it implies dividing the number by 1 which doesn’t do anything to the numbers value, but is very important when dealing with negative fractions. if you show the hidden fraction in the previous example it would look like , and like this it makes more sense why we got the answer from earlier if we had to first get the reciprocal then evaluate from there.

This also works “backwards” in a way, if you had to rewrite with a positive exponent it would just turn into as you flip the fraction and don’t need to show the hidden 1 under the exponent now.

In a more complex question like where there is already a fraction, the only movement happening because of the negative exponent would only be the number (or variable in this case) that has the negative exponent attached to it, moving the negative exponent to the top half of the fraction gets us where we can then carry onto regular math with variables, (positive) exponents and fractions, where the answer would be .\

That’s a summary of what I’m confident in being able to explain about negative exponents from the week I’ve spent learning about them. Thanks for reading.