Why Does the Order of Square/Square Root Matter?

Date: 8/28/96 at 15:40:32
From: Ken Mead
Subject: Why Does the Order of Square/Square Root Matter?

Why does the square root of x^2 equal absolute value of x, but the 
square of the square root of x equals just plain old x?

Sweating in Precalc,  Ken

Date: 8/28/96 at 22:33:1
From: Doctor Robert
Subject: Re: Why Does the Order of Square/Square Root Matter?

It is mostly a matter of semantics, but when you see a square root 
sign, which I will abbreviate sqrt(   ), the answer is, by agreement, 
a positive number.  sqrt(9) = 3 because sqrt(3*3) = 3.  

If you are asked sqrt(-3*-3) you might expect to write -3, but by 
agreement, answers are only positive, therefore you must write 
abs(-3)=3.   

When you put x^2 under the radical, you don't know whether x will be 
positive or negative; therefore you must write sqrt(x^2) = abs(x).  

I hope that this helps.

-Doctor Robert,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/