Powers, Roots and Indices
Index Notation
An index (or power/exponent) tells you how many times a base is multiplied by itself.
$$a^n = a \times a \times a \times \cdots \times a \quad (n \text{ times})$$
Examples:
- $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$
- $3^4 = 81$
- $10^6 = 1{,}000{,}000$
Laws of Indices
| Law | Rule | Example |
|---|---|---|
| Multiplication | $a^m \times a^n = a^{m+n}$ | $x^3 \times x^4 = x^7$ |
| Division | $a^m \div a^n = a^{m-n}$ | $y^8 \div y^3 = y^5$ |
| Power of a power | $(a^m)^n = a^{mn}$ | $(2^3)^2 = 2^6 = 64$ |
| Zero index | $a^0 = 1$ | $7^0 = 1$ |
| Negative index | $a^{-n} = \frac{1}{a^n}$ | $3^{-2} = \frac{1}{9}$ |
Fractional Indices
A fractional index indicates a root: $$a^{1/n} = \sqrt[n]{a}$$ $$a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$$
Examples:
- $25^{1/2} = \sqrt{25} = 5$
- $8^{1/3} = \sqrt[3]{8} = 2$
- $4^{3/2} = (\sqrt{4})^3 = 2^3 = 8$
- $27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9$
Key tip: Always find the root first, then apply the power — this keeps numbers smaller.
Square and Cube Roots
The square root of $a$ is the number that multiplies by itself to give $a$: $$\sqrt{a} \times \sqrt{a} = a$$
The cube root of $a$ is the number that multiplies by itself three times to give $a$: $$\sqrt[3]{a} \times \sqrt[3]{a} \times \sqrt[3]{a} = a$$
Key squares to know: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Key cubes to know: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
Negative and Fractional Index Summary
| Expression | Meaning | Value (if $a=4$) |
|---|---|---|
| $a^0$ | 1 | 1 |
| $a^{-1}$ | $\frac{1}{a}$ | $\frac{1}{4}$ |
| $a^{-2}$ | $\frac{1}{a^2}$ | $\frac{1}{16}$ |
| $a^{1/2}$ | $\sqrt{a}$ | 2 |
| $a^{3/2}$ | $(\sqrt{a})^3$ | 8 |
WJEC Exam Tips
- On non-calculator papers: you must recall squares up to 15² and cubes up to 10³.
- Show all steps when simplifying indices — method marks are available.
- For fractional indices on higher tier: always state which root you are taking.
- Negative indices: $5^{-3} = \frac{1}{5^3} = \frac{1}{125}$ — do not confuse with making the number negative.
✦Worked example
Evaluate $64^{2/3}$
Step 1: Identify the root: denominator is 3, so take the cube root of 64. $$\sqrt[3]{64} = 4$$
Step 2: Apply the power from the numerator: raise to the power 2. $$4^2 = 16$$
Answer: $64^{2/3} = 16$
AI-generated · claude-opus-4-7 · v3-wjec-maths