



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: 





5^{0}






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? 






2^{0} 
= 
1 




6^{0} 
= 
1 




321,587^{0} 
= 
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: 





4^{2}






It means: 





4^{2} = 4
× 4 =
16 





If we have: 





4^{0}






It means: 





4^{0} =
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
3^{4}






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: 






3^{4} =  (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 
