Pre Algebra Special Tricks With Exponents
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Even Fancier Multiplying
Special Tricks With Exponents

 
 Check this out:
 

 (42)3

 
 What do you think this one means? 
 Well, just take it one step at a time:
 

 (42)3 = 42 42 42

 
 And we know that 4 2 = 4 x 4, so:
 

 (42)3 = 42 42 42 = 4 4 4 4 4 4 = 46 = 4,096

 
 Let's try something else. WATCH CLOSELY:
 

 52 + 3 = 52 53     NOT      52 + 53

 
 Here's why:
 

 52 + 3 = 55 = 5 5 5 5 5

 
 Now let's do some grouping with parenthesis:
 

 55 = (5 5) (5 5 5)

 (Since all of the signs are "x" we don't change anything by grouping)

 
 The next step is to write each of these groups as a number with an exponent:
 

 55 = (5 5) (5 5 5) = 52 53

 
 And there you go!

 

 OK, How about this:
 

 23 33

 
 Can we combine this one?
 
 Let's see. First, break it down:
 

 23 33

 

 2 2 2 3 3 3

 
 Now mix up the numbers a bit:
 

 2 3 2 3 2 3

 
 Next, put in some parenthesis in special places:
 

 (2 3) (2 3) (2 3)

 
 Multiply the stuff inside each parenthesis:
 

 6 6 6

 
 And write this as a number and an exponent:
 

 63

 
 OK, Big Deal, So What, Who Cares?
 
 Here's the deal.
 
 We started with:
 

 23 33

 
 And ended with:
 

 63

 
 That means:
 
IF THE EXPONENTS ARE THE SAME 
 IN A MULTIPLICATION PROBLEM, 
 YOU CAN JUST MULTIPLY THE NUMBERS UNDER THEM! 
 

 23 33 = (2 3)3 = 63 

 
 Examples:
 

 75 45 = (7 4)5 = 285

 

 26 46 = (2 4)6 = 86

 

 3-2 4-2 = (3 4)-2 = 12-2

 
 What happens with something like:
 

 35 + 75

 
 Can we leave the exponents in place and add this thing together somehow?
 

 NO!

 
 If you read the chapters up to here, you probably remember problems like:
 

 30 = 1

 

 
 Back then, I said that you just had to accept that as so. 
 I said that sometime later I would show you why it works that way.
 
 IT'S TIME!
 
 At the start of this chapter we had stuff like this:
 

 52 53 = 5(2 + 3) = 55

 
 Check out this one:
 

 53 5-3

 
 We can deal with this one in two ways. The first way is:
 

 53 5-3 = 5(3 - 3) = 50 = 1

 
 Oh yeah? So what? That doesn't prove ANYTHING!
 
 That's true, but now look at the second way we can do this one:
 

 
 Now put a denominator of 1 under the 5 3 and we have:
 

 
 Now multiply and simplify:
 

 
 And there it is!
 

 53 5-3 = 50 = 1

 

   copyright 2005 Bruce Kirkpatrick

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