National 5 Mathematics: expressions and formulae

Expressions and formulae is the algebra half of National 5 Maths, and most of it is sat without a calculator — so the marks turn on whether the manipulation is automatic. This set drills the ten things that come up over and over: simplifying a surd, the index laws, a fractional index, expanding a pair of brackets, factorising a trinomial and a difference of two squares, cancelling an algebraic fraction, changing the subject of a formula, rounding to significant figures, and substituting a negative value.

Every wrong option is a slip candidates actually make rather than a random number — a sign lost when the middle terms are collected, the root and the power applied in the wrong order, decimal places counted instead of significant figures, a negative squared and left negative. The explanations show the working line by line and name which specific mistake produces the closest wrong answer, so a wrong click tells you what went wrong and not just that something did.

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• Simplify a surd by pulling out the largest square factor
• Apply the index laws for multiplying, dividing and raising a power to a power
• Evaluate a fractional index by taking the root first and then the power
• Expand a pair of brackets and collect the middle terms without losing a sign
• Factorise a trinomial and recognise a difference of two squares
• Simplify an algebraic fraction by factorising and cancelling
• Change the subject of a formula using inverse operations in the right order
• Round to significant figures, counting from the first non-zero digit
• Substitute negative values correctly, keeping a squared negative positive

Evaluate $8^{\frac{2}{3}}$ as a whole number. — The denominator 3 is the root and the numerator 2 is the power, so $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$.