The goal of this exercise is to make sure you understand √x^2 = |x| and in general√x^2 ̸= x.

Question

a) Check your textbook (course notes) to find the fact that for any real number a such that a ≥ 0, there is a unique number √a ≥0. Which fact is that?

b) Evaluate the following (write them without square root): sqrt{(−5)^2} = −−−−−, sqrt{5^2} = −−−−−, If a > 0 : sqrt{a^2} = −−−−− , sqrt{(−a)^2} = −−−−−−.

Do b.

Answer 1

square root of x^2 = sqr(x^2) = |x|

because the root sign means only the positive square root of x^2 = not x as x may be <0,

sqr(x^2) = the positve square root only which is the same as the absolute value of x = |x|. It's a matter of definitions of math symbols.

if sqr(x^2) = x, and x were negative that would be an impossible contradiction, as sqr(x^2) is only the positive square root of x