Fractional Exponent Rule

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:

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. \[ 9^{\frac{1}{2}} = \sqrt{9} = 3 \]
2. \[ 27^{\frac{1}{3}} = \sqrt[3]{27} = 3 \]
3. \[ 8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4 \]
4. \[ 16^{\frac{3}{4}} = \sqrt[4]{16^3} = \sqrt[4]{4096} = 8 \]

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}