Integrated Math 2 / Unit 1: Exponents and Radicals
1.3 Rational Exponents
Notes
A rational exponent is a fraction exponent. It links powers and roots.
- $\displaystyle a^{1/n} = \sqrt[n]{a}$, the $n$th root of $a$.
- $\displaystyle a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$. The denominator is the root and the numerator is the power.
- Example: $\displaystyle 8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$.
- A negative rational exponent means a reciprocal: $\displaystyle 16^{-1/2} = \frac{1}{4}$.
- Take the root first, since it keeps the numbers small.
Practice
What is $64^{1/3}$?
The exponent $\displaystyle \frac{1}{3}$ means cube root, and $4^3 = 64$.
Evaluate $8^{2/3}$.
Cube root of 8 is 2, then square it: $2^2 = 4$.
Evaluate $16^{-3/4}$.
Fourth root of 16 is 2, cubed is 8, and the negative sign gives the reciprocal $\displaystyle \frac{1}{8}$.
Ready for more? Take the Unit 1 test.