sqrt{ a^2 }

College Algebra
Section R.8

Given an example to show that sqrt{a^2} is not equal to a. Use it to explain why sqrt{a^2} equals the absolute value of a.

This is the last review section problem. This weekend we move on to College Algebra material beginning with Chapter 1.

 

It is the definition of square root of a number. The square root is defined in the sense that sqrt(a^2)=|a| for all real a.
Thus, the domain is the real numbers and the codomain is the non-negative real numbers.

The reason it is defined this way is to make sure that f is a function. Assume for a minute that
sgrt(a^2)=a, then for example:

sqrt((−5)^2)=−5, sqrt(5^2)=5
But we know that sqrt((−5)^2)=sqrt(25)= sqrt(5^2). Thus we see that sqrt(25)=−5,5.
And thus f(a^2) is not a function. To keep it as a function, we have to 'sacrifice' and say that f(a^2)≠a. Rather, f(a^2)=|a|. This will be consistent with the definition of a function.

 

I never knew this fact about square roots. Of course, if we are taking the square root of a number in terms of distance, negative distance is rejected.

I found this unanswered thread from Sunday afternoon:

Simplify Algebraic Expression...2

Did I simplify the algebraic expression correctly?