fractional exponents

Question

Can you give a clear explanation as to when fractional exponents can be reduced (or expanded)? For example (8)^(1/3) does not equal (8)^(2/6), as this second item has 6 roots (some of them complex) and the first item has only 3 roots. So by reducing 2/6 to 1/3 roots are eliminated. My notation (8)^(1/3) means (8) raised to the (1/3) power.

Answer 1

8^(2/6) = root6( 8*8) = root6( 2*2*2*2*2*2) = 2 = root3(8)

So in the REAL CASE,

If the fractions are equal then the expressions must also hold by substitution

Now for the imaginary case:

(-8)^(2/6) = root6[ (-1 * 2*2*2)*(-1 * 2*2*2) ] =

root6[ 2^6] = 2

If the fraction can be reduced, then there has to be common factor of 2,3,5 or some other prime.

For odd exponent factors, the negative will remain.

For even exponents, the imaginary will cancel