Pre Algebra Special Exponents
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Fancier Multiplying
Special Exponents

 
 OK, before we go any farther with exponents
 there are two special cases we need to talk about. 
 
 The first special case is where the exponent is zero. 
 Like in:
 

 50

 
 The value of this is something you probably wouldn't have guessed. 
 The rules say that ANY NUMBER "to the zero power" equals, 
 get this, ONE! 
 
 Too weird eh?
 
  20 = 1
  60 = 1
  321,5870 = 1
 
 I know, I know. 
 EVERYBODY says "but stuff like that should equal zero not one!"
 Well what can I tell you, it equals 1 and that's the way it is. 
 
 Sometimes things in math come up 
 that at the time you just have to accept and go on. 
 This is one of them. 
 (Later on, after we do a bunch more exponent stuff, 
 we'll come back to this one and see why it works this way.)
 
 The other special case is when we have exponents 
 that are negative numbers.
 Something like:

 

 4-3

 
 What does this puppy mean? 
 Before you even try to guess, I'll tell you. 
 The negative sign just means the number and the exponent 
 go in the denominator (bottom part) of a fraction. 
 And if there is nothing else around a 1 goes in the numerator (top part).
 
 So:
 

 
 STOP! HOLD ON! TIME OUT!
 
 Look at this stuff again. If we have:
 

42 

 
 It means:
 

42 = 4 4 = 16

 
 If we have:
 

 40

 
 It means:
 

 40 = 1

 
 If we have:
 

 4-1

 
 It means:
 

 
 EVEN WHEN THE EXPONENT IS A NEGATIVE NUMBER, 
 THE ANSWER WE GET IS POSITIVE
 
 OK, it's a number really close to zero, but it's still positive.
 
 OK.
 
 POP QUIZ TIME. True or false
 

 4-5,000,000 is greater than 0

 

 TRUE!

 
 Hey, it's not much greater than zero, 
 but even a bit bigger than zero is enough.
 
 Got it? Great, let's go on.
 Since we've been talking about negative stuff, what about something like this:
 

 (-6)5

 
 This one is no special case or anything. 
 It's just -6 times itself five times. 
 It's:
 

 (-6) (-6) (-6) (-6) (-6) = -7776

 
 So (-6)5 = -7776. 
 You may have figured this next part out already, 
 but I'll tell you anyway. 
 
 If you have a negative number "raised to an even number power" 
 like:
 

 (-3)4 = 81

 
 You get a positive number for an answer.
 

 BUT

 
 If the exponent is an odd number, like:
 

 (-3)5 = -243

 
 You get a negative number for an answer.
 Let's put it all together.
 How about something like this:
 

 (-5)-4

 
 OK, no big thing, just take it step by step. 
 The exponent has a negative sign so put the whole thing in the denominator:
 
 So:
 

 
 Now forget about the rest of the problem for a minute
 and just work on the denominator:
 

 (-5)4 is just (-5) (-5) (-5) (-5) = 625

 (the exponent is an even number so the answer is positive)

 
 So putting the problem all together:
 

 
 The exponent in that problem was an even number 
 so the answer comes out positive.
 
 The minus sign on the exponent 
 means all the "action" happens in the denominator.
 
 Did you notice that when we raised negative numbers to powers
 we always use parenthesis?
 
 We write:
 

 (-3)4 and not -34

 
 WHY?
 
 Because in the list of what order you do things, 
 exponents come before +, -, x, !
 
 So with parenthesis we get:
 

 (-3)4 = (-3) (-3) (-3) (-3) = 81

 
 And without parenthesis we get:
 

 - 34 = - (3 3 3 3) = - 81

 
 Get it?
 We now have a more complete list of the order you do things in. 
 The order is:
 

 Parenthesis

 

 Exponents

 

 Multiplication and Division

 

 Addition and Subtraction

 
 One way to remember the order these go in
 is to make a sentence from the first letters of the words. 
 
  Parenthesis
  Exponents
  Multiplication
  Division
  Addition
  Subtraction
 
 A famous one of these for this is ...
 
  Please
  Excuse
  My
  Dear 
  Aunt 
  Sally
 
 But if you don't like that one you can make up your own.
 

   copyright 2005 Bruce Kirkpatrick

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