Powers, roots and surds

• Calculate with squares, square roots, cubes, cube roots and other powers and roots.
• Recall squares from 1² to 15² and the cubes of 1, 2, 3, 4, 5 and 10 without a calculator.
• Simplify surds: √20 = 2√5 and √200 − √32 = 6√2.
• Multiply and expand expressions containing surds.
• Rationalise the denominator: 10/√5 = 2√5 and 1/(−1 + √3) = (1 + √3)/2.

Recall these without a calculator

Cubes to know: 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 10³ = 1000. So ∛64 = 4 and ∛1000 = 10. The cube root of a negative number is negative: ∛(−8) = −2.

Use the power and root keys, with brackets

On Paper 2, use the x^□ key for powers and the ^□√ key for roots. Put a whole expression under a root sign inside brackets, and write down more figures than you need before rounding.

Take out the largest square factor

A surd is an irrational root such as √5, written exactly. Use √(ab) = √a × √b: split the number into a square number times something else, and take the square root of the square.

Multiply every term, then tidy

Expand exactly as in algebra. Every √a × √a becomes a, so surd terms often cancel.

Remove the root from the bottom

To rationalise the denominator means to rewrite the fraction with no surd in the denominator.