Structure and calculation Revision Notes

Nearby known values

For a square root, find consecutive squares surrounding the target; for a cube root, find consecutive cubes. The required root must lie between the corresponding integer roots. The closer the target is to one known power, the closer the estimate should be to that root.

Interpolation and checking

A useful estimate can be made by comparing the target's position between two nearby powers. For example, 82 lies just above 81, so its square root is just above 9. Check the result by raising the estimate to the relevant power and confirming that it is close to the original positive number.

Powers and fractional powers

For powers such as a number squared, cubed or raised to a fractional power, use nearby familiar calculations. A fractional power may represent a root followed by a power, so compare with convenient values such as 4 squared, 4 cubed or 16 to the power one-quarter. State an appropriate approximation rather than implying an exact value.

Adding and subtracting multiples of pi

Write each term as a fraction multiplied by π. Find a common denominator for the numerical fractions, combine the numerators, and retain π. For example, 3π/4 + π/2 becomes 3π/4 + 2π/4 = 5π/4.

Multiplying and dividing exactly

Multiply the fractional coefficients and leave π attached unless another π is present. When dividing by a fraction, multiply by its reciprocal. For example, (2π/3) ÷ (4/5) = (2π/3) × (5/4) = 5π/6. Check that the final fraction is in simplest form.

Comparing exact answers

To compare positive multiples of π, compare their fractional coefficients, since both contain the same positive factor π. Use a common denominator where useful. For instance, 5π/4 = 15π/12 and 7π/6 = 14π/12, so 5π/4 is greater.

Simplifying surds

Rewrite the number inside a square root as a product containing the largest possible square factor. For example, √72 = √(36 × 2) = 6√2. Only like surds can be added or subtracted, so 3√5 + 2√5 = 5√5, whereas √2 and √3 cannot be combined.

Rationalising denominators

For a denominator of √a, multiply top and bottom by √a, so c/√a becomes c√a/a. For a denominator such as a + √b, multiply by the conjugate a − √b. The difference-of-squares result makes the denominator rational: (a + √b)(a − √b) = a² − b.

Reliable exact-value method

Simplify every surd before collecting terms, then rationalise only the final denominator. Check that no square factor remains inside a surd and that the denominator contains no root. Decimal approximations can be used as a rough check, but the final answer should remain exact.

Core method

Identify every stage of the process and count the options available at each stage. Multiply these numbers together. If there are a choices at stage one, b at stage two and c at stage three, the total number of outcomes is a × b × c.

Checking your answer

Make sure each outcome contains exactly one choice from every stage and check whether repetition is allowed. A code may allow repeated digits even when selecting physical objects would not. Estimate the size of the answer and recalculate the multiplication to reduce errors.