The Fractional Exponent Rule states that an exponent in the form of a fraction represents a combination of roots and powers. It follows this general form:

$$a^{\frac{m}{n}} = \left( \sqrt[n]{a} \right)^m = \sqrt[n]{a^m}$$

Breakdown:

1. The denominator ( n ) of the fraction represents the root (i.e., the nth root of the base).
2. The numerator ( m ) represents the power (i.e., raising the result to the mth power).

Examples:

1. Square root example

   $$9^{\frac{1}{2}} = \sqrt{9} = 3$$

2. Cube root example

   $$27^{\frac{1}{3}} = \sqrt[3]{27} = 3$$

3. Combination of root and power

   $$8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$$

4. Another way to compute

   $$16^{\frac{3}{4}} = \sqrt[4]{16^3} = \sqrt[4]{4096} = 8$$

This rule helps simplify complex expressions involving roots and exponents, making calculations more systematic.

Latex

a^{\frac{m}{n}} = \left( \sqrt[n]{a} \right)^m = \sqrt[n]{a^m}