



Once
upon a time



you might
have learned about things called roots. 





Maybe you
called it a radical. 





Maybe you
just called it a rad. 





Whatever
you called it, 


it looked
like this ... 











It
doesn't mean much by itself. 


It needs
to have something inside of it ... 











All
together, the number and the squiggle means: 





THE
NUMBER THAT TIMES ITSELF EQUALS 4






That is
... 











Now we
just have to figure out 


what
number times itself equals 4. 





This one
us not too tough. 


Most
people know that ... 


2
x 2 = 4






That
means: 











(At
the end of this page, we will talk about another answer to this
problem. 


To be
completely correct, we should mention this answer now 


Later
on in Algebra, we will use this other answer in some problems) 





OK, but
what's this got to do with X? 





Thanks, I
was hoping that you would ask ... 


What if
you have this problem ... 





X^{2}
= 25






How do
you turn this puppy into: 





X
= SOMETHING






We can't
get there by adding or subtracting anything. 


We can't
get there by multiplying or dividing by anything. 


What now? 











So what
we do here, 


is to
drop one of those root things onto each side of the = . 


Like this
... 


X^{2}
= 25












So what
have we got? 





What does
this mean? 











It means
the number that times itself equals 25. 


So what
number times itself equals 25? 





5 × 5 = 25






So: 











That
makes our problem ... 











Now lets
look at the messy X part. 


What does
this mean? 











It means
... 





THE
THING THAT TIMES ITSELF EQUALS X^{2}






So what
times itself equals X ^{2}? 





X × X = X^{2}






So: 











That
makes our problem ... 

















OK,
here's that little detail I mentioned earlier. 





The extra
answer you get in these problems 


works
like this. 





When we
said 


"What
times itself equals 4?" 





We gave
the answer as 2. 


That's
true. But there is another number that times itself equals 4. 





The other
answer is 2. 





A
negative times a negative is a positive. 


So 2 x
2 = 4 also. 





Later on,
we will use this "negative root" in problems. 


For now,
we will mostly ignore it. 





The
positive root that we will use, 


is most
often called the principal square root. 





copyright 2005 Bruce Kirkpatrick 
