How do you evaluate fractional exponents?

#x^(a/b) =rootb(x^a) = (rootb(x))^a#

You can just remember this rule, or you can learn about why this is:

fractional exponent #1/b#

So first we're going to look at an expression of the form: #x^(1/b)#.
To investigate what this means, we need to go from #x to x^(1/b)# and then deduce something from it.

#x^1 = x^(b/b) = x^(1/b*b)#
What does multiplication mean? Repeated addition. So we can instead of multiplying by b, adding the number to itself #b# times.
#x^(1/b+1/b+1/b+1/b +...)# (b times)

There is a rule you use when multiplying numbers with the same radical: add the exponents. If we reverse this rule, we get:
#x^(1/b)*x^(1/b)*x^(1/b)*x^(1/b)*x^(1/b)...# (b times)

Now, we still know that this number is equal to #x#. So now we have to think a bit. What number, multiplied by itself b times, gives you #x#.
It's the bth-root of #x# => #x^(1/b)=rootbx#

For example: #8^(1/3)#
If we multiply this by itself 3 times we get:
#8^(1/3)*8^(1/3)*8^(1/3) = 8^(3/3) = 8#
What number multiplied by itself 3 times, gives you 8.
It's of course #root3(8) = 2#

What about #a/b#?
To know what #x^(a/b)# means, we can further rely on our previous findings:
#x^(a/b) = x^(a*1/b) = x^(1/b+1/b+1/b+1/b...) # (a times)
#= x^(1/b)*x^(1/b)*x^(1/b)...# (a times)

Repeated multiplication is equal to exponentiation, so we can write:
#= (x^(1/b))^a = (rootbx)^a#

You can also bring the exponent in the root:
#= rootb(x^a)#