AP Precalculus: Rational Exponents Formulas & Rules

1. Evaluate Rational Exponents

• \( a^{m/n} = (\sqrt[n]{a})^{m} = \sqrt[n]{a^m} \)
• Example: \( 8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4 \)
• \( a^{1/n} = \sqrt[n]{a} \)
• \( a^{-k/n} = 1/(\sqrt[n]{a^k}) \)

2. Convertible Forms

• \( a^{m/n} \leftrightarrow \sqrt[n]{a^m} \)
• Example: \( \sqrt[5]{x^2} = x^{2/5} \)

3. Operations and Properties

• \( a^r \cdot a^s = a^{r+s} \)
• \( \frac{a^r}{a^s} = a^{r-s} \)
• \( (a^r)^s = a^{r\cdot s} \)
• \( (ab)^r = a^r b^r \)
• \( \left(\frac{a}{b}\right)^r = \frac{a^r}{b^r} \)
• Negative exponent: \( a^{-r} = \frac{1}{a^r} \)

4. Simplifying Expressions with Rational Exponents

• Use all laws above to combine/reduce
• Factor out roots/powers: \( a^{1/2}b^{3/2} = (\sqrt{a})(b \sqrt{b}) \)
• Write all as single base exponent when possible

5. Solving Equations with Rational Exponents

• Raise both sides to reciprocal power: \( x^{m/n}=a \implies x=(a)^{n/m} \)
• Check solutions: must satisfy domain restrictions of original radical (no negatives for even roots)