Number - Structure and calculation: calculate exactly with fractions, surds and multiples of π; simplify surd expressions involving squares (e.g. √12 = √(4 × 3) = √4 × √3 = 2√3) and rationalise denominators (Higher tier)

The idea

Higher tier only. Exact calculation keeps a value in fraction, surd or π form instead of rounding. Fractions: 2/3 + 1/4 = 8/12 + 3/12 = 11/12, and 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8. Multiples of π: the circumference of a circle radius 3 cm is 2π × 3 = 6π cm, left as 6π rather than 18.8. A surd is a root that is not exact, such as √2. Simplify by splitting off the largest square factor: √50 = √25 × √2 = 5√2. Like surds combine: 3√2 + 4√2 = 7√2, but √2 + √3 cannot. Products: √6 × √3 = √18 = 3√2. Rationalise by multiplying top and bottom by the root: 6/√3 = 6√3/3 = 2√3. For a two-term denominator such as 1 + √5, multiply by the conjugate 1 − √5, since the difference of two squares removes the root.

What earns the marks

• Calculating exactly with fractions, for example 2/3 + 1/4 = 11/12, without converting to a rounded decimal.
• Leaving an answer as a multiple of π, such as 6π for the circumference of a circle of radius 3 cm, rather than a decimal.
• Splitting a surd at its largest square factor, such as writing √50 as √25 × √2 = 5√2.
• Multiplying numerator and denominator by the same root, or by the conjugate of a two-term denominator, to rationalise.
• A fully simplified exact answer, with no root left in the denominator.

Common mistakes and how to put them right

• Writing √a + √b as √(a + b), which already fails for √9 + √16, since that is 7 and not 5.

• Multiplying only the denominator by the root when rationalising, which changes the value of the fraction.

• Rounding a multiple of π or a surd partway through, so the final answer is no longer exact.

• Changing the sign of only one term when multiplying by the conjugate, so the roots fail to cancel.

Exam tips

• Learn the square numbers as far as 144 so you can spot the largest square factor quickly.
• Keep fractions, surds and π exact the whole way through; rounding partway loses the accuracy mark.
• After rationalising, check that no root is left in the denominator before writing the final line.