Integrated Math 3 / Unit 4: Radical Functions and Equations
4.1 Roots and Rational Exponents
Notes
The $n$th root undoes the $n$th power: $\displaystyle \sqrt[n]{a} = a^{1/n}$. A rational exponent combines a root and a power: $\displaystyle a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$.
- $\displaystyle 8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$.
- $\displaystyle 16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$.
- A negative exponent flips the base: $\displaystyle 27^{-1/3} = \frac{1}{\sqrt[3]{27}} = \frac{1}{3}$.
- Simplify radicals by pulling out perfect squares: $\displaystyle \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.
Practice
Evaluate $8^{2/3}$.
Take the cube root first: $\displaystyle \sqrt[3]{8} = 2$, then square it.
Evaluate $27^{-1/3}$.
The negative exponent gives the reciprocal, and $27^{1/3} = 3$.
Simplify $\displaystyle \sqrt{50}$.
$\displaystyle \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.
Ready for more? Take the Unit 4 test.