Launching your learning experience…
Exam Season Offer 🎯 Sign up today & get 15% OFF on Yearly Plan
Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next. (2026)
All rights reserved
©2026 Tutopiya
All resources on this platform are independently created by Tutopiya and have no endorsement from the International Baccalaureate Organization.
Question
Show that the HCF of 84 and 126 is 42, and find the LCM.
Solution
Write each number as a product of prime factors.
HCF: take each prime common to both, using the lower power.
LCM: take every prime, using the higher power.
Check: HCF × LCM should equal 84 × 126.
Answer
, .
Examiner note
AQA mark schemes award one mark for each correct prime factorisation, one mark for the HCF and one mark for the LCM. The check at step 4 is not required but catches errors instantly.
Question
Show all steps to evaluate , leaving your answer as a fraction.
Solution
Deal with the negative sign by taking the reciprocal.
Apply the fractional index: root first (cube root), then power (square).
Substitute back.
Answer
.
Examiner note
The two most common errors: (1) raising to the power before rooting, giving which is impossible to cube root by inspection; (2) forgetting to apply the negative index and leaving the answer as 25.
Question
Write in the form , where and are integers.
Solution
Identify the conjugate of the denominator: .
Multiply numerator and denominator by the conjugate.
Expand the denominator using the difference of two squares.
Expand the numerator.
Write the final answer.
Answer
, so , .
Examiner note
One mark is typically awarded for identifying the conjugate, one for expanding the denominator correctly to a rational number, and one for the final simplified form.
Question
Give your answer in standard form.
Solution
Divide the coefficients.
Divide the powers of 10 by subtracting the indices.
Combine. The coefficient 3 satisfies , so no adjustment needed.
Answer
.
Examiner note
The most common error is instead of . Subtracting a negative exponent adds, not subtracts.
Index law — multiplication
When to use
When multiplying two powers that have the same base — add the indices. Do not apply when the bases are different.
Example
.
Index law — division
When to use
When dividing two powers with the same base — subtract the index of the denominator from the index of the numerator.
Example
.
Index law — power of a power
When to use
When a power is raised to another power — multiply the indices.
Example
.
Negative index
When to use
Any time you see a negative index — rewrite as a reciprocal first, then evaluate the positive power.
Example
.
Fractional index
When to use
Whenever the index is a fraction. Take the th root first to keep numbers manageable on Paper 1.
Example
.
A positive integer greater than 1 that has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13. Note: 1 is not prime.
Related: prime factorisation, factor
Expressing a positive integer as a product of prime numbers. The unique factorisation theorem states there is exactly one such product for every integer greater than 1 (ignoring order). Written in index form: .
The largest positive integer that divides exactly into two or more given integers. Found by multiplying the prime factors common to all numbers, using the lowest power of each.
because , , shared factors: .
Related: prime factorisation, LCM
The smallest positive integer that is a multiple of two or more given integers. Found by multiplying all prime factors, using the highest power of each.
because all prime factors with highest powers: .
Related: prime factorisation, HCF
An irrational root that cannot be simplified to a rational number. , , , are surds; is not.
Related: rationalise the denominator, irrational number
Rewrite a fraction so that the denominator contains no surds. For a simple surd denominator , multiply numerator and denominator by . For a binomial denominator , multiply by the conjugate .
A way of writing very large or very small numbers as , where and is an integer. for large numbers, for small numbers.
; .
Mistake
Using the higher power for HCF or the lower power for LCM.
Why it happens
Students confuse which rule applies to HCF and which to LCM.
How to avoid it
Remember: HCF = Humble (takes the smaller power, shared factors only). LCM = Large (takes the larger power, all factors).
Source: AQA Examiner's Report — Number topic, repeated across series.
Mistake
Raising to the power before taking the root for fractional indices: computing then trying to cube root it.
Why it happens
Students apply the index mechanically left-to-right.
How to avoid it
For : root first (), then power. The numbers stay small enough to handle without a calculator.
Mistake
.
Why it happens
Distributing the square root over addition, the same way you might with multiplication.
How to avoid it
Prove the error with a counterexample: . The square root distributes over multiplication only.
Mistake
Leaving a coefficient outside , e.g. writing instead of .
Why it happens
Students perform the arithmetic correctly but forget to check the standard form condition on the coefficient.
How to avoid it
After every standard-form calculation, ask: 'Is my coefficient between 1 and 10?' If not, adjust: move the decimal one place and change the power by 1.
Mistake
Always doing multiplication before division in BIDMAS.
Why it happens
The M in BIDMAS comes before D, so students assume multiplication has higher priority.
How to avoid it
Division and multiplication are equal priority — evaluate left to right. , not .