Exponents and radicals are inverses -- like addition and subtraction or multiplication and division. Exponents and radicals "undo" each other. Exponent of radical of Z = Z in the same way that Z + 5 - 5 = Z or (5 x Z) / 5 = Z. This is true for both whole numbers and rational numbers. Rational numbers are not numbers that make more sense than other numbers. They are numbers that are written as ratios, such as 3/4 or 1/2. The rules for exponents and radicals involving rational numbers are exactly the sameas the rules involving whole numbers.
All exponents obey the same three properties (whether the exponents are whole numbers or rational numbers).
Property # 1: When the same number (called the base) has two different exponents, the product of the exponentiated numbers is the base with the sum of the exponents. An example will make it clearer: (B^A) x (B^C) = (B^A + C).
Property # 2: When the exponentiated numbers are divided the new exponent is calculated by subtraction. Example: (B^A) / (B^C) = (B^A - C).
Property # 3: When the exponentiated number is exponentiated the resulting exponentation is a product. Example: ((B^A)^C)) = (B^(A x C)).
An example of property # 1 using rational numbers: (10^3/4) x (10^5/4) = (10^(3/4 + 5/4)) = (10^8/4) = (10^2) = 10 x 10 = 100.
The exponentation by A/B is undone by the radical A/B, but it is also undone by the exponentation by B/A. This is true because or property # 3: ((B^A/B)^B/A) = (B^(A/B x B/A)) = (B^1) = B. Therefore exponentation by B/A undoes exponentation by A/B. This means that we could get by without using radicals at all -- we could just use exponentation, and this is actually a common practice. To see how this works, consider the radical expression for the square root of a number (the number that is multiplied by itself to get a number -- the square root of 49 is 7 because 7 x 7 = 49). In radical notation (49 sqrt 2) = 7. But we could also say (49^1/2) = 7. Here is why this works: (49^1/2) x (49^1/2) = (49^(1/2 + 1/2() = (49^1) = 49.