fractional exponent

[vicktor schausberger]

what is the demostration about √A=A ^{1/2
always they try to explain by multiply A½ x A½ why?}

 

[HallsofIvy]

There is, strictly speaking, no "demonstration", \(\displaystyle A^{1/2}= \sqrt{A}\) is the definition of \(\displaystyle A^{1/2}\).

We can give an indication of why that is a good definition:

If we define "\(\displaystyle A^n\) to be A(A)(A)....(A), that is, A mutiplied by itself n times, then we can show that \(\displaystyle (A^n)^m= A^{nm}\) so that, in particular, if we want that to be true even for fractions, like 1/2, we have to have \(\displaystyle (A^2)^{1/2}= A^{(1/2)(2)}= A\). That is, the 1/2 power has to be the "reverse" of \(\displaystyle A^2\), the square root of A.

 

[Bob Brown MSEE]

I agree with Hall's post.
However I have seen the square root sign used to indicate the positive value only.

 

[HallsofIvy]

That is the way I was intending it! Both \(\displaystyle A^{1/2}\) and \(\displaystyle \sqrt{A}\) indicate the positive number whose square is A.