Powers, roots and indices
Powers and roots
- a power (index) shows repeated multiplication: $2^5 = 32$.
- a square root reverses squaring ($\sqrt{169} = 13$); a cube root reverses cubing ($\sqrt[3]{8} = 2$).
- Example: $5^2 \times \sqrt[3]{8} = 25 \times 2 = 50$.
Practice
Work out 5² × ∛8.
5² = 25 and ∛8 = 2, so 25 × 2 = 50.
The laws of indices
$$a^m \times a^n = a^{m+n}, \qquad a^m \div a^n = a^{m-n}, \qquad (a^m)^n = a^{mn}$$
- Special powers: $a^0 = 1$, $a^{-n} = \dfrac{1}{a^n}$, $a^{1/n} = \sqrt[n]{a}$ (fractional powers are Extended).
Practice
Using the index laws, what is 2³ × 2⁴ as a single number?
2³ × 2⁴ = 2⁷ = 128.
Practice
Evaluate 81^(1/2).
81^(1/2) = √81 = 9.
You've got it
Key idea
- multiply powers → add indices; divide → subtract; power of a power → multiply
- $a^0 = 1$, $a^{-n} = \dfrac{1}{a^n}$, $a^{1/n} = \sqrt[n]{a}$
- $5^2 \times \sqrt[3]{8} = 50$