Simplifying the Square Root of a Sum

Date: 09/27/2004 at 18:36:20
From: Alyssa
Subject: Why does sqrt(a + b) not equal the sqrt(a) + sqrt(b)

Why is it that sqrt(a + b) does not equal sqrt(a) + sqrt(b)?

I.e., the sqrt(36 + 49) = sqrt(85) or about 9.2195.  But the
sqrt(36) + sqrt(49) = 13.

This confuses me because I know that the sqrt(ab) = sqrt(a)sqrt(b).  I
don't understand why under the operations of division or
multiplication the sqrt can be separated out but in the operations of
addition or subtraction it can not.

Date: 09/27/2004 at 19:01:08
From: Doctor Schwa
Subject: Re: Why does sqrt(a+b) not equal the sqrt(a) + sqrt(b)

Hi Alyssa,

Maybe it would help to think of things this way:

  a * (b + c) = a * b + a * c 

for all numbers, because multiplication is repeated addition.  If 
a = 3, you could write out 3*(b+c) = b+c + b+c + b+c and then simplify
to get 3b + 3c because it's all addition.

  Similarly, (a * b)^c = a^c * b^c 

because exponents are repeated multiplication.  If c = 3, you could
write out (a*b)^3 = a*b * a*b * a*b and then simplify to get a^3 * b^3
because it's all multiplication.

  But (a + b)^c is NOT the same as a^c + b^c 

because exponents aren't repeated addition.  If c = 3, you could write
out (a+b)^3 = (a+b) * (a+b) * (a+b) but then you can't simplify the *
and the + mixture!

In much the same way, sqrt can be thought of as an exponent (it's the
1/2 power), so 

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

but 

  sqrt(a+b) can't be rearranged in any nice way.

Does that help it make more sense?

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