(√x)^2=|x| ? ( if we square the root, will we get the module ( (√x)^2=|x| )?)

[politexnik]

Good afternoon, could you tell if we square the root, will we get the module ( (√x)^2=|x| )?

 

[stapel]

Good afternoon, could you tell if we square the root, will we get the module ( (√x)^2=|x| )?

Reverse the order:

[imath]\qquad \sqrt{x^2\;} = \vert x\vert[/imath]

To be sure, plug in a negative value; say, [imath]-9[/imath]. Your formulation says:

[imath]\qquad \left(\sqrt{-9\;}\right)^2 = \left(3i\right)^2 = -9[/imath]

The other order, however, says:

[imath]\qquad \sqrt{(-9)^2\;} = \sqrt{81\;} = +9[/imath]

Which matches the absolute value?

 

[The Highlander]

Good afternoon, could you tell if we square the root, will we get the module modulus ((√x)^2=|x| )?

Yes, the radix or radical sign \(\displaystyle \left(\sqrt{ }\right)\) is defined as the principal square root so it is always positive.
Therefore, \(\displaystyle \sqrt{x^2\;} = \vert x\vert\) is true.

As explained in the first section of this article.
NB: Pay particular attention to the two sentences I have highlighted for you. ?
(You can remove the highlighting by simple refreshing the page.)

Hope that helps. ?

 

[Bruce]

principal square root is always positive.

There's one exception: The principal square root of zero (sometimes described as the "trivial case", but still worth mentioning).