Powers, roots, integer indices and standard form
Index notation and the laws of indices
When a number is multiplied by itself repeatedly, we use index notation. For example, 2 × 2 × 2 = 2³ (read "2 to the power 3"). The number 2 is the base and 3 is the index (or exponent or power).
The five laws of indices
| Law | Rule | Example |
|---|---|---|
| Multiplication | aᵐ × aⁿ = aᵐ⁺ⁿ | x³ × x⁴ = x⁷ |
| Division | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | x⁶ ÷ x² = x⁴ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (x²)³ = x⁶ |
| Zero index | a⁰ = 1 | 7⁰ = 1 |
| Negative index | a⁻ⁿ = 1/aⁿ | x⁻² = 1/x² |
Fractional indices (Higher tier):
- a^(1/n) = ⁿ√a (the nth root)
- a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)
Example: 8^(2/3) = (∛8)² = 2² = 4.
Square and cube roots
Square root (√): the number that, when multiplied by itself, gives the original number. √25 = 5 because 5² = 25. Note: both +5 and −5 are square roots of 25, but √25 conventionally means the positive root.
Cube root (∛): the number that, when cubed, gives the original. ∛27 = 3.
Key square roots to know: √4 = 2, √9 = 3, √16 = 4, √25 = 5, √36 = 6, √49 = 7, √64 = 8, √81 = 9, √100 = 10, √121 = 11, √144 = 12, √169 = 13, √196 = 14, √225 = 15.
Key cube roots: ∛8 = 2, ∛27 = 3, ∛64 = 4, ∛125 = 5, ∛1000 = 10.
Standard form (standard index form)
Standard form expresses very large or very small numbers as A × 10ⁿ where 1 ≤ A < 10 and n is an integer.
Large numbers: move the decimal point left to find A; the number of moves is n. Example: 45,000 = 4.5 × 10⁴ (decimal moved 4 places left).
Small numbers: move the decimal point right; n is negative. Example: 0.0072 = 7.2 × 10⁻³ (decimal moved 3 places right).
Calculations in standard form (non-calculator):
- Multiply: multiply the A values, add the exponents. If A > 10, adjust.
- Divide: divide the A values, subtract the exponents.
Example: (3 × 10⁵) × (4 × 10³) = 12 × 10⁸ = 1.2 × 10⁹.
⚠Common mistakes— Common mistakes (CCEA examiner traps)
- Forgetting that a⁰ = 1 for any non-zero a. Students write a⁰ = 0.
- Confusing negative index with negative number: 2⁻³ = 1/8, not −8.
- Standard form with A ≥ 10 or A < 1: 12 × 10³ is NOT standard form — adjust to 1.2 × 10⁴.
- Fractional indices: confuse numerator and denominator. In a^(m/n), the denominator n is the root and the numerator m is the power.
- Multiplying powers wrongly: x² × x³ = x⁵ (add exponents), NOT x⁶.
AI-generated · claude-opus-4-7 · v3-ccea-maths