An exponent is just shorthand for multiplying a number times itself. The value of the exponent is just a rule for the number of times the number is to be multiplied by itself. So, "x" raised to the "a" power is x multiplied by x multiplied by x..."a" times. One way of writing exponents in text is as x^a, which is just the same as x raised to the a power. As an example, 2^5 is 2*2*2*2*2, which is 32. It doesn't matter if the exponent is small or large, a whole number or a fraction --- it's the same rule. But what does it mean to multiply a number times itself 1/2 times?
The nature of exponents leads to a few rules. One rule that leads to an explanation for how to interpret a fractional exponent is the multiplication rule for exponents. The multiplication rule is simple: x^a times x^b is x^(a+b). Consider x^(1/2) times x^(1/2). Using the multiplication rule, this equals x^(1/2 + 1/2) which is x^1, which is x times itself only one time, which is x. So x^(1/2) * x^(1/2) = x. So x^(1/2) is something that, when squared, will equal x. That is, x^(1/2) times itself is x.
The function that meets that criterion is the square root. So x^(1/2) is the same as sqrt(x). Using the same procedure, you can see that x^(1/3) is the cube root of x, x^(1/9) is the ninth root of x, and x^(1/a) is the "a"th root of x. More complicated functions are easy to deal with; you just do both operations. For example, x^(5/3) is two operations together: taking the cube root of x, and multiplying x by itself 5 times. The operations can be done in either order. So, 4^(3/2), for example, is either (sqrt(4))^3 or sqrt(4^3). Both give the same answer: 2^3 is 8, and sqrt(64) is 8.
Even more complex expressions can be easily handled. For example, x * x^(5/3) * x^(5/12) can be rewritten as x^(1 + 5/3 + 5/12). The addition can be done using the standard rules for adding fractions finding the lowest common denominator. The example becomes x^(12/12 + 20/12 + 5/12), which is x^37/12. And, although it may be a strange-looking calculation, this is the same as the 12th root of (x^37) or the (12th root of x)^37.