Question

If sqrt(x ^ 2) = |x| and (sqrt(x))^2 = x, why does the fractional exponent rule says x^(a/2) = sqrt(x^a) = (sqrt(x))^a?

Answer 1

Because x^1/2 is only defined for nonnegative real x regardless of any real a.

Answer 2

first it is important to note for what sets of real numbers each statement holds

sqrt(x^2 ) = |x| is true for all x in R

(sqrt(x))^2 = x is true for all x>=0

since in real number calculations the sqrt function is undefined for negative inputs

however if we extend the definition of the function with complex numbers then for x<0:

(sqrt(x))^2

= (sqrt(-|x|))^2

= (sqrt(|x|)* i)^2

= i^2 |x|

= -|x|

= x

hence it overall follows for all x in R: (sqrt(x))^2 = x

now about your second statement:

x^(a/2) = sqrt(x^a )

this is certainly true for x,a>=0

it also holds for a<0 and x=/=0

if x<0 then issues start to appear, for example for x=-1 and a=2

(-1)^(2/2) = -1 but

sqrt((-1)^2 ) = sqrt(1) = 1

i hope that answers your question