



Check this
out: 





(4^{2})^{3}






What do you think this one means? 


Well, just take it one
step at a time: 





(4^{2})^{3}
= 4^{2}
×
4^{2}
×
4^{2}






And we know that 4
^{2 } = 4 x 4,
so: 





(4^{2})^{3} =
4^{2}
×
4^{2}
×
4^{2} =
4 ×
4 × 4 ×
4 × 4 ×
4 = 4^{6} =
4,096






Let's try something else. WATCH
CLOSELY: 





5^{2 +
3} = 5^{2}
×
5^{3}
NOT 5^{2} +
5^{3}






Here's why: 





5^{2 +
3} = 5^{5} = 5
×
5 ×
5 ×
5 ×
5






Now let's do some grouping with
parenthesis: 





5^{5} = (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: 





5^{5} = (5
×
5) ×
(5 ×
5 ×
5) =
5^{2}
×
5^{3}






And there you go! 





OK, How about
this: 





2^{3}
×
3^{3}






Can we combine this one? 





Let's see. First, break it down: 





2^{3}
×
3^{3}






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: 





6^{3}






OK, Big Deal, So What, Who Cares? 





Here's the deal. 





We started with: 





2^{3}
×
3^{3}






And ended with: 





6^{3}






That means: 





IF THE EXPONENTS ARE THE
SAME 


IN A MULTIPLICATION PROBLEM, 


YOU CAN JUST MULTIPLY THE NUMBERS UNDER
THEM! 





2^{3}
×
3^{3} = (2
×
3)^{3} =
6^{3}






Examples: 





7^{5}
×
4^{5} = (7
×
4)^{5} =
28^{5}






2^{6}
×
4^{6} = (2
×
4)^{6} =
8^{6}






3^{2}
×
4^{2} = (3
×
4)^{2} =
12^{2}






What happens with something like: 





3^{5} +
7^{5}






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: 





3^{0} =
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: 





5^{2}
×
5^{3}
= 5^{(2 + 3)}
=
5^{5}






Check out this one: 





5^{3}
×
5^{3}






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





5^{3}
×
5^{3} = 5^{(3  3)} =
5^{0} = 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! 





5^{3}
×
5^{3}
= 5^{0}
=
1






copyright 2005 Bruce Kirkpatrick 
