Indices and Standard Form
Indices (powers)
An index (plural indices), or power, shows how many times a number is multiplied by itself: 2⁴ = 2 × 2 × 2 × 2 = 16. The small raised number is the index.
The laws of indices
| Law | Rule | Example |
|---|---|---|
| Multiplying | aᵐ × aⁿ = a^(m+n) — add the powers | 2³ × 2⁴ = 2⁷ |
| Dividing | aᵐ ÷ aⁿ = a^(m−n) — subtract the powers | 5⁶ ÷ 5² = 5⁴ |
| Power of a power | (aᵐ)ⁿ = a^(m×n) — multiply the powers | (3²)⁴ = 3⁸ |
| Power of 0 | a⁰ = 1 | 7⁰ = 1 |
| Negative power | a⁻ⁿ = 1 ÷ aⁿ | 2⁻³ = 1/8 |
| Fractional power | a^(1/n) = ⁿ√a | 9^(1/2) = √9 = 3 |
(The multiply/divide laws only work when the base is the same.)
Standard form
Standard form writes very large or very small numbers compactly as:
A × 10ⁿ where 1 ≤ A < 10, and n is a whole number- Large numbers: positive n. 4 500 000 = 4.5 × 10⁶.
- Small numbers: negative n. 0.0003 = 3 × 10⁻⁴.
The power of 10 = how many places the decimal point moves (right for positive, left for negative).
Calculating with standard form
- Multiplying: multiply the numbers, add the powers.
(3 × 10⁴) × (2 × 10³) = 6 × 10⁷. - Dividing: divide the numbers, subtract the powers.
- Adding/subtracting: convert to ordinary numbers (or the same power of 10) first.
- Always adjust so A is between 1 and 10 at the end.
Worked example
Simplify (2 × 10⁵) × (4 × 10³).
- Numbers: 2 × 4 = 8. Powers: 10⁵ × 10³ = 10⁸.
- Answer: 8 × 10⁸. ✓
Common mistakes
- Multiplying the powers when you should add them (for aᵐ × aⁿ).
- Writing standard form with A not between 1 and 10 (e.g. 45 × 10⁵ is wrong — should be 4.5 × 10⁶).
- Getting the sign of the power wrong for small numbers (should be negative).
Exam tips
- Learn the index laws precisely — a very common topic.
- Check standard form: the first number must be ≥ 1 and < 10.
- For calculations, deal with the numbers and the powers of 10 separately.
Key facts to remember
- Index laws: × → add powers, ÷ → subtract, power of a power → multiply; a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^(1/n) = ⁿ√a.
- Standard form: A × 10ⁿ with 1 ≤ A < 10; positive n for large numbers, negative for small.
- Multiply/divide in standard form by handling the numbers and powers of 10 separately.