Algebra 2 / Unit 4: Radical Functions and Rational Exponents
4.2 Rational Exponents
Notes
A rational exponent is a fraction exponent: $\displaystyle a^{\frac{1}{n}} = \sqrt[n]{a}$ and $\displaystyle a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$.
- The denominator is the root; the numerator is the power. Taking the root first keeps numbers small.
- Example: $\displaystyle 27^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9$.
- A negative exponent means reciprocal: $\displaystyle a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}}$.
- The usual exponent rules still work: $a^m \cdot a^n = a^{m + n}$, $(a^m)^n = a^{mn}$.
Practice
Evaluate $\displaystyle 8^{\frac{2}{3}}$.
$\displaystyle \sqrt[3]{8} = 2$ and $2^2 = 4$.
Evaluate $\displaystyle 16^{-\frac{3}{4}}$.
$\displaystyle 16^{\frac{3}{4}} = 2^3 = 8$, and the negative exponent gives the reciprocal.
Simplify $\displaystyle x^{\frac{1}{2}} \cdot x^{\frac{1}{3}}$.
Add exponents: $\displaystyle \frac{1}{2} + \frac{1}{3} = \frac{5}{6}$.
Ready for more? Take the Unit 4 test.