sqrt{ a^2 }

[nycmathguy]

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.

 

[MathLover1]

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.

 

[nycmathguy]

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?