This was the question:

I'm having trouble with exponents. I don't understand what you mean by 'to the 4th power' or something like that. My dad said it had something to do with being able to write a number in shorter form. That and something about making a number smaller or larger. Could you help me?
-Confused

Professor Homunculus' answer:

Any number (let's call it "n") to any power (let's call the power "x") is simply the number times itself x times.

An example would be:

3 to the 4th power would be 3 times itself 4 times. That means 3 x 3 x 3 x 3.

Be careful, and don't mistake that for 3 x 4. It is very different. 3 x 4 is 12.

3 x 3 x 3 x 3 is 81. Big difference. (By the way, thanks to the astute reader who corrected this. Originally I had this answer as 729. Professor Homuculus stands corrected!)

A good rule of thumb, is when you see "n" to the "x" power, write the n down x times, and put multiplication signs between them. In the above case, we wrote the number 3 down 4 times, and put multiplication signs between them.

One more rule is that any number raised to the first power is the original number itself. For example, 3 to the first power is 3, 5 to the first power is 5, 19 to the first power is 19, and so on.

Numbers raised to powers can grow very large very quickly. (Maybe that's why your dad recognized that you can write a number in a shorter form using exponents.) For example, you normally write "Two to the tenth power" as the number 2, with a small 10 written above it and to the right. On a computer with a simple font, in which you can't make the little ten appear higher, you have to write it as 2^10. Either way, it is a lot easier than writing 1,024 (which is 2^10, or 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2.

Imagine how long 10^33 would look if you had to write it out. It would be the number 1 with 33 zeroes behind it!

You might also want to know that the number you are raising to a power is called the "root." And the power you are raising it to is called the "exponent". So in 6^3, 6 would be the root, and 3 would be the exponent. If you worked out 6^3 (6 x 6 x 6) you would get 216. You would say, "Six to the third power equals 216." You could also say, "The third root of two hundred sixteen is six."

A great, inexpensive, easy-to-understand book that explains exponents very well, is Rosza Peters' "Playing with Infinity" (available from Dover Press - any bookstore should be able to order it easily for you, or you can go to the homepage of this website and click on the “Great Math Books” link in the navigation bar, and order it directly from there). It is a "must" math book, as far as I am concerned.

I hope this helped!

Hi-Ho!

Professor Homunculus