Here x ² > 0 for all real numbers x. Therefore the equation  x ² = b only has solutions x when b > 0, that is only when b is non-negative.  Defining

b^½ =sqrt(b)

as the nonnegative real solution of  x ² = b works only  if b is positive. This solution is given by a ^½ = exp( ½ln(b)). See above.

Similarly, if n = 2m > 0 is an even, then x ⁿ = x ^2m > 0 for all real numbers x. So   the equation  x ^2m = b only has solutions x when b > 0, that is only when b is non-negative. The foregoing implies defining

b^½m =as the 2m root of (b)

as the nonnegative real solution of  x ^2m = b works only  if b is positive.  This solution is then given by a^1/n = exp( (1/n)ln(b)) where n = 2m.  See above

Related posts:

1. Natural Logarithms and Exponentials – Roots and Powers
2. Exponentials of Real Numbers a x = exp( x ln(a))