



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 ... 





3^{8} =
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: 





3^{3}
× 2^{3} = (3 × 2)^{3} = 6^{3} = 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 
