Rational Exponents

Definition

A rational exponent is a fraction exponent:

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

Special Cases

$a^{\frac{1}{n}} = \sqrt[n]{a}$

Examples:

• $x^{\frac{1}{2}} = \sqrt{x}$
• $x^{\frac{1}{3}} = \sqrt[3]{x}$
• $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$

General Form

$a^{\frac{m}{n}}$

• Numerator ($m$): power
• Denominator ($n$): root

Example: $27^{\frac{2}{3}}$

Method 1: $27^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 3^2 = 9$

Method 2: $27^{\frac{2}{3}} = \sqrt[3]{27^2} = \sqrt[3]{729} = 9$

Negative Rational Exponents

$a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}}$

Example: $16^{-\frac{3}{4}} = \frac{1}{16^{\frac{3}{4}}} = \frac{1}{(\sqrt[4]{16})^3} = \frac{1}{2^3} = \frac{1}{8}$$

All Exponent Rules Apply!

• Product: $a^m \cdot a^n = a^{m+n}$
• Quotient: $\frac{a^m}{a^n} = a^{m-n}$
• Power: $(a^m)^n = a^{mn}$