Five to the Third PowerDate: 09/20/2001 at 17:17:03 From: Daniel Subject: Powers/exponents Dear Dr. Math, The one thing that is giving me most trouble is powers and exponents, such as: 5 to the 3rd power is? On a test I put 15. Wrong! Help, Daniel Date: 09/21/2001 at 14:30:53 From: Doctor Ian Subject: Re: Powers/exponents Hi Daniel, Have you learned about prime factoring yet? For example, 36 has lots of factors: 72 = 1 * 72 36 = 2 * 36 = 3 * 24 = 4 * 18 = 6 * 12 = 8 * 9 But it can be reduced to a set of prime factors: 72 = 2 * 2 * 2 * 3 * 3 and these factors can be used to produce all the other factors: 2 * 2 = 4 2 * 3 = 6 2 * 2 * 2 = 8 2 * 2 * 3 = 12 2 * 2 * 3 = 18 2 * 2 * 2 * 3 = 24 2 * 2 * 3 * 3 = 36 Now, it's not very convenient to write 72 = 2 * 2 * 2 * 3 * 3 so it's somewhat natural to look around for a more compact notation. One such notation would be to count the number of times that a prime factor appears, and use that number instead of repeating the factor that many times. Using this notation, 3 2 72 = 2 * 3 because 2 appears three times, and 3 appears twice. Let's look at how another factorization might be written: 2 4 2 * 3 * 4 * 4 * 5 * 5 * 5 * 5 = 2 * 3 * 4 * 5 Note that when something appears only once, it gets no exponent, although we _could_ use the exponent 1: 1 1 2 4 = 2 * 3 * 4 * 5 But that makes more work, not less. Note also that when you get larger and larger collections of the same factor, this can really save a lot of writing. For example, would you rather write 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 or 13 2 ? It looks as though, during your test, you saw the 5 and the 3 together and thought you were supposed to multiply. And that's a very natural thing to think, if you aren't familiar with the notation for exponents. But where 5 * 4 would be read '5 times 4', and indicates that you're supposed to multiply 5 by 4, 4 5 would be read '5 to the 4th power', and indicates that you're supposed to multiply 5 by 5 four times in a row. Once you've got the notation down, the next step is to get a handle on the basic properties of exponents. You can find a gentle introduction to them here: Properties of Exponents http://mathforum.org/dr.math/problems/crystal2.01.22.01.html I hope this helps. Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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