Powers and roots

Powers and roots appear across all three OCR J560 papers. The index laws are essential algebra tools; knowing square and cube numbers by heart saves time under exam conditions.

Powers (indices)

aⁿ means a multiplied by itself n times.

• 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.
• 10³ = 1000.
• 3⁴ = 81.

Any number to the power of 0 equals 1: a⁰ = 1 (for a ≠ 0). Any number to the power of 1 equals itself: a¹ = a.

Index laws

For the same base:

Law	Rule	Example
Multiply	aᵐ × aⁿ = aᵐ⁺ⁿ	3⁴ × 3² = 3⁶
Divide	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	5⁶ ÷ 5² = 5⁴
Power of a power	(aᵐ)ⁿ = aᵐⁿ	(2³)⁴ = 2¹²

Negative indices: a⁻ⁿ = 1/aⁿ. Example: 2⁻³ = 1/2³ = 1/8.

Square numbers and square roots

Know squares 1² to 15² (and ideally to 20²): 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.

Square root (√) undoes squaring: √81 = 9 because 9² = 81.

Every positive number has two square roots: +√n and −√n. Example: √64 = 8 but also −8 (though we usually take the positive root).

Cube numbers and cube roots

Know cubes 1³ to 5³: 1, 8, 27, 64, 125.

Also useful: 6³ = 216, 10³ = 1000.

Cube root (∛) undoes cubing: ∛125 = 5 because 5³ = 125.

Cube roots of negative numbers exist: ∛(−8) = −2.

Estimating roots

For non-perfect squares, find the nearest perfect squares on each side.

Example: √50. Since 7² = 49 and 8² = 64, √50 is between 7 and 8, closer to 7.

Example: √50 ≈ 7.07 (calculator check: 7.07² = 49.98 ✓).

Common OCR exam mistakes

1. Confusing 2³ with 2 × 3 = 6. The correct answer is 8.
2. Applying index laws across addition: 3² + 3² ≠ 3⁴. You can only use index laws when multiplying or dividing the same base.
3. Thinking negative numbers have no square root — they don't have REAL square roots (at GCSE level), but they do have cube roots.
4. Forgetting the negative root when solving x² = 25: x = ±5, not just 5.