Rational exponent explain why the square root of a number is defined to be equal to that number to the 1/2 power.

The concept of radicals and radical exponents is tricky at first, but makes sense when we look into the logic behind it. 

When we write a radical in exponential form, like writing √x as x^(1/2), we are simply putting the power of the radical in the denominator (bottom number) of the exponent, and the numerator is the power we raise the exponent to, or the power that would be inside the radical. 
In our example, √x is really ²√(x¹), or the square root of x to the first power. For this reason, we write it as x^(1/2).

Let's say we wanted to write the cubed root of x squared, in exponential form. 
In radical form, it would look like this:
³√(x²) . This means we square x, and then take the cubed root.
In exponential form, remember that we take the power of the radical (3), and make that the denominator of the exponent, and keep the numerator as the power that x is raised to (2). 
Therefore, it would be x^(2/3), or x to the 2 thirds power. 

Just like when multiplying by a fraction, you multiply by the numerator and divide by the denominator, in exponential form, you raise your base number to the power of the numerator, and take the root of the denominator.