This was the question:

Can you explain how to understand negative exponenents, like 5^-4 ?

Professor Homunculus' answer:

Negative Exponents

Here is the explanation why 5^-4 equals 1/5⁴ (or 1/625). In other words, why a whole number raised to a negative power can be simplified into a fraction raised to a positive power.

To understand this lesson, you will have to understand the basics of exponents ("powers"). You can get those basics at: http://mathmojo.com/interestinglessons/intro_exponents/Intro_exponents.html

You must also understand the order of operations. Luckily, that is the most recent lesson posted at mathmojo.com. If you need a refresher, the direct link to it is:
http://mathmojo.com/interestinglessons/order_of_operations/order_of_operations.html

You must also understand that in order to divide exponents with the same base, you simply subtract the powers. To give you a quick reason why, take this example: 2⁶/2⁴.
2⁶= 2*2*2*2*2*2 (that's 6 groups of 2), and 2⁴ = 2*2*2*2 (that's 4 groups of 2).
When you cancel four groups of 2 from both the numerator and the denominator (see below), you end up with 2*2 which is 2². So 2⁶/2⁴ = 2².

In the case of 4¹/ 4⁴ = 4^-3 you would subtract 4 from 1 and get -3, giving you 4^-3.

Therefore,  if 4^-3 is the same as 4¹/ 4⁴, we know that that is the same as 4/4*4*4*4. Now, if you cancel out a four in the denominator, and a four in the numerator, you end up with 1/4*4*4 which is the same as 1/4³ (or 1/64).

In other words:
A number raised to a negative power, is the same the same as its reciprocal raised to the same power, only positive.

Try that with 5^-4 and see if you get the same thing as in the first sentence, way up at the top of this lesson.

Then, for the "fun" of it, please simplify:

3^-3
8^-3
6^-4

and just to be cruel, try

(x^-3)^-4 (Don't forget the order of operations!)

Have fun!

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