Imaginary number manipulations

[Mr Davis 97]

I know that this is probably a very commonly asked question with students, but say that we have $\sqrt{(-1)^2}$. If we performed the innermost operation first, then we have $\sqrt{(-1)^2} = \sqrt{1} = 1$. However, according to rules for radicals, we can do $\sqrt{(-1)^2} = (\sqrt{-1})^2 = -1$. Which one of these is the correct way of going about evaluating the expression, and why?

 

[member 587159]

I know that this is probably a very commonly asked question with students, but say that we have $\sqrt{(-1)^2}$. If we performed the innermost operation first, then we have $\sqrt{(-1)^2} = \sqrt{1} = 1$. However, according to rules for radicals, we can do $\sqrt{(-1)^2} = (\sqrt{-1})^2 = -1$. Which one of these is the correct way of going about evaluating the expression, and why?

$\sqrt{(a)^2}$ = |a|

 

[FactChecker]

If you are really going to study imaginary and complex numbers, you will find out that roots do not have unique values, but rather they can have several "branches". Notice that (-1)² = 1. So you could also have a square root, sqrt(), of 1 where sqrt( -1² ) = sqrt( 1) = -1. The standard definition of √ is that it is the "principle branch". But that is just a convention, not a mathematical rule.