Square Root

Date: 03/15/99 at 09:45:55
From: Sean Mooney
Subject: Square Root

Why cannot you take the square root of a sum or difference separately -
i.e. why is sqrt(9+4) not equal to sqrt(9) + sqrt(4)?

Date: 03/15/99 at 17:48:35
From: Doctor Rick
Subject: Re: Square Root

You want to know why the square root of 13 is not sqrt(9) + sqrt(4), 
which is 3+2 = 5. If 5 were the square root of 13, then we could 
square 5 and we would get 13. Do we? No, we get 25. So clearly it does 
not work. But why not?

Let us look into this a little more closely. We will square 
(sqrt(9) + sqrt(4)) using FOIL (or the distributive property):

  (sqrt(9) + sqrt(4))^2 = sqrt(9)^2 + 2*sqrt(9)*sqrt(4) + sqrt(4)^2
                        = 9         + 2*sqrt(36)        + 4

You see that we do get 9 and 4, but we also get a "cross" term, 
2*sqrt(36). It is this cross term that messes up the pattern.

What you are asking about, or perhaps wishing for, is a sort of 
distributive property of square root over addition, that works the 
same way the usual distributive property (of multiplication over 
addition) works:

     c(a+b) = c*a + c*b           (TRUE - distributive property)

  sqrt(a+b) = sqrt(a) + sqrt(b)   (FALSE)

You can distribute a product over the members of a sum, but you cannot
distribute a square root over the members of a sum. However, you CAN
distribute a square root over the members of a PRODUCT:

  sqrt(a*b) = sqrt(a) * sqrt(b)   (TRUE)

This is a special case of a distributive property of POWERS over
MULTIPLICATION.

       n    n    n
  (a*b)  = a  * b

If n = 1/2, this is the same as above.

You can think in terms of a hierarchy of operations:

  powers/roots
  multiplication/division
  addition/subtraction

Think about this: powers are to multiplication as multiplication is to
addition. When you first learned multiplication, it was defined as 
adding a number to itself a certain number of times: 3 * 5 = 5 + 5 + 5. 
In the same way, powers were first defined as multiplying a number by 
itself a certain number of times: 5^3 = 5 * 5 * 5. When you left the 
realm of integers, these simple definitions were no longer enough; but 
logarithms put the analogy on a solid mathematical footing for real 
numbers, transforming multiplication into addition and powers into 
multiplication.

The distributive property of powers over multiplication follows from 
the definition of powers in the same way that the distributive property 
of multiplication over addition follows from the definition of 
multiplication. But there is no comparable way of deriving a property 
for powers and addition; you cannot jump 2 steps in the hierarchy and 
get a distributive property.

I hope this answers your question, but I also hope it stirs up more
questions in your mind. There are a lot of interrelationships in the 
world of numbers waiting to be explored.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/