Pre Algebra Roots II
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More Radical!
Roots II

 
 Let's say you had a nasty problem like this:
 

 

 
 Now maybe you know that:
 

 
 But with that exponent of 8 in there, things are not so simple.
 
 A really nice trick with exponents and roots 
 is that when you have stuff like this,
 
 you can do either the exponent first or the root first. 
 Do first whichever makes the problem easier. 
 It's your choice.
 

 
 So we can do ...
 

 
 and then do ...
 

 38 = 6,561

 
 Yeah, you probably need a calculator for that last part (at least I did!).
 
 This little trick works both ways. 
 If we have:

 

 

 
 We can do the exponent first:
 

 

 
 Some people don't like the looks of the radical squiggle. 
 They think it looks too much like a division problem. 
 They wanted another way to write these.
 
 Here it is. 
 A plain radical means the same as an exponent of one half. 
 That is:
 

 
 Radicals with numbers in them are the same as fraction exponents
 with that number in the denominator (bottom part).
 

 

 
 AND ... Watch closely:
 

 
 SAY WHAT?
 

 

 
 Or, saying it another way:
 

 

 
 So if we had:
 

 

 
 With this fraction exponent trick 
 we can use all the rules we came up with for exponents. 
 Like the one about combining things with the same exponent.
 
 If you read the exponent pages, 
 you might remember problems like this:
 

 33 23 = (3 2)3 = 63 = 6 6 6 = 216

 
 When we change roots to exponent fractions, we can do that too!
 
 So, Change:
 

 

 
 To:
 

 
 2 x 8 = 16, so we have:
 

 
 And this is still just a square root, so we can say: 
 "What times itself equals 16?
 
 The answers are 4 and - 4.
 

   copyright 2005 Bruce Kirkpatrick

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