OxfordAQA International GCSE Mathematics: Index laws, surds, standard form and sets (9260) – Revision Notes

These notes are aligned to the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1 (for exams May/June 2018 onwards), statements N6 to N9 in section 3.1.1. Core content is also assessable on the Extension tier; Extension-only items are labelled. For full explanations, read the study guide. Then test yourself with the practice questions.

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Rules at a glance

Ref	Rule or definition	Tier
N6	aᵐ × aⁿ = aᵐ⁺ⁿ	Core
N6	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	Core
N6	a⁰ = 1 and a⁻ⁿ = 1/aⁿ	Core
N6	a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ	Extension only
N7	Fractions: common denominator to add or subtract; multiply across; divide by multiplying by the reciprocal	Core
N7	√(ab) = √a × √b and √(a/b) = √a / √b	Extension only
N7	(a + √b)(a − √b) = a² − b	Extension only
N8	Standard form: a value from 1 up to 10, times a whole-number power of 10	Core
N9	Count of A ∪ B = count of A + count of B − count of the overlap	Core

N6: Index laws

Method for mixed terms

1. Handle the numbers (coefficients) on their own.
2. Handle each letter separately: add indices to multiply, subtract to divide.
3. If the answer must have positive indices, move each negative-index factor into the denominator.

Reminder: 4a³ × 6a⁻⁷ = 24a⁻⁴ = 24/a⁴. The coefficients multiply (4 × 6) but the indices only add (3 + (−7) = −4).

Must-know distinctions

• 2⁻³ = 1/8 (a small positive number), not −8 and not −6.
• 7⁰ = 1, not 0.
• (5/3)⁻² means flip then square: (3/5)² = 9/25.
• The laws need the same base. 2⁴ × 3⁴ cannot be combined by adding indices.
• To solve equations such as 2^(x+1) = 1/16, write both sides as powers of the same base, then equate the indices: x + 1 = −4, so x = −5.

Fractional powers (Extension only). Take the root first, then the power. Reminder: 32^(2/5): 32^(1/5) = 2, and 2² = 4. A negative fractional power is a reciprocal as well: 4^(−3/2) = 1/(4^(3/2)) = 1/8.

N7: Exact calculation

Fractions: method in steps

1. Convert each mixed number to an improper fraction.
2. Adding or subtracting needs a common denominator. Multiplying or dividing is easier if you cancel first.
3. To divide, invert the second fraction and multiply.
4. Give the final answer as a fraction (simplified), as a mixed number if the question suggests it, and never as a rounded decimal when “exact” is asked for.

Reminder: 1½ × 2⅔ = 3/2 × 8/3 = 4. Reminder: 2/3 ÷ 4/9 = 2/3 × 9/4 = 3/2.

Surds (Extension only): method in steps

1. Simplify each surd by taking out the largest square factor (√45 = √9 × √5 = 3√5).
2. Combine only like surds: 3√5 + 2√5 = 5√5.
3. When expanding, remember √a × √a = a.
4. To rationalise, multiply top and bottom by the surd (single surd) or by the conjugate (two-term denominator).

Denominator	Multiply top and bottom by	Result
√6	√6	denominator 6
4 + √3	4 − √3	denominator 13
5 − √2	5 + √2	denominator 23

Reminder: 9/√6 = 9√6/6 = 3√6/2. Cancel after rationalising.

Reminder for a two-term denominator: 8/(4 + √3) = 8(4 − √3)/13 = (32 − 8√3)/13. There is nothing to cancel here, because 13 does not divide 32 and 8.

N8: Standard form

Converting

• Move the decimal point until you have a number from 1 up to (not including) 10.
• Count the places moved. Large number: positive power. Number between 0 and 1: negative power.
• Reminder: 0.00805 = 8.05 × 10⁻³ and 3.7 × 10⁶ = 3 700 000.

Calculating

• Multiply or divide: do the A values, then the powers of 10, then tidy A back into the range 1 to 10.
• Add or subtract: make the powers of 10 equal first. The powers themselves do not get added.
• Reminder: (6 × 10⁴) × (5 × 10⁻⁷) = 30 × 10⁻³ = 3 × 10⁻².

Calculator displays. A display such as 2.4E-5 means 2.4 × 10⁻⁵. In your answer, write the × and the power of 10 in full.

Worked reminder (add and subtract). (5.2 × 10⁵) − (3 × 10⁴): write 3 × 10⁴ as 0.3 × 10⁵. Then 5.2 × 10⁵ − 0.3 × 10⁵ = 4.9 × 10⁵. For a small-number case, (2 × 10⁻³) + (5 × 10⁻⁴) = 2 × 10⁻³ + 0.5 × 10⁻³ = 2.5 × 10⁻³.

Comparing sizes. Compare the powers of 10 first. If the powers are equal, compare the A parts. For example, 8.1 × 10⁻³ is smaller than 2 × 10⁻², because −3 < −2.

N9: Sets

Notation	Reading	How to find it
ξ	universal set	the full list or the rectangle
A′	not in A	everything in ξ outside circle A
A ∪ B	A or B (or both)	all regions in either circle
A ∩ B	A and B	the overlap only
n(A)	number of elements in A	count the elements, or add the region totals

Method for Venn diagram problems

1. Draw ξ as a rectangle and a circle for each set. Overlap circles even if the overlap may be empty.
2. Fill the central region first (all sets), then the pairs-only regions by subtraction, then the single-set regions.
3. The outside region is the total minus everything inside the circles.
4. Check that all regions add up to n(ξ).

Reminder for two sets: if 30 pupils study Spanish or French, 18 study Spanish and 16 study French, then both = 18 + 16 − 30 = 4.

Worked reminder (three sets). In a group of 30, 12 take art, 10 take music, 7 take drama. 3 take art and music only, and 2 take all three. Nobody takes drama and music without art, and 4 take art and drama only. Then art only = 12 − 3 − 2 − 4 = 3, music only = 10 − 3 − 2 = 5, and drama only = 7 − 4 − 2 = 1. Inside the circles: 3 + 5 + 1 + 3 + 4 + 2 + 0 = 18, so 30 − 18 = 12 take none of them.

Describing regions. (A ∪ B)′ is the outside region only. A′ ∩ B is “in B but not in A”, which is the part of circle B outside the overlap.

Quick self-test

Try these without looking at the answers. Questions 4 to 7 are Extension only.

1. Simplify 4a³ × 6a⁻⁷, giving your answer with a positive index.
2. Find the value of 9⁰ + 2⁻¹.
3. Simplify x⁹ ÷ x⁻².
4. (Extension only) Evaluate 32^(2/5).
5. (Extension only) Simplify √45 + √20.
6. (Extension only) Rationalise 9/√6 and simplify.
7. (Extension only) Expand and simplify (3 − √2)(3 + √2).
8. Calculate 1½ × 2⅔.
9. Write 0.00805 in standard form.
10. Calculate (6 × 10⁴) × (5 × 10⁻⁷), giving your answer in standard form.
11. Calculate (8.4 × 10⁶) ÷ (2 × 10⁻²), giving your answer in standard form.
12. ξ = {1, 2, …, 10}, A = {1, 3, 5, 7, 9}, B = {2, 3, 5, 7}. (a) List A ∩ B. (b) Find n(A ∪ B). (c) List (A ∪ B)′.

Answers

1. 24a⁻⁴ = 24/a⁴
2. 1 + 1/2 = 3/2 (or 1.5)
3. 9 − (−2) = x¹¹
4. 32^(1/5) = 2, then 2² = 4
5. 3√5 + 2√5 = 5√5
6. 9√6/6 = 3√6/2
7. 9 − 2 = 7
8. 3/2 × 8/3 = 4
9. 8.05 × 10⁻³
10. 30 × 10⁻³ = 3 × 10⁻²
11. 4.2 × 10⁶⁻⁽⁻²⁾ = 4.2 × 10⁸
12. (a) {3, 5, 7}. (b) A ∪ B = {1, 2, 3, 5, 7, 9}, so 6. (c) {4, 6, 8, 10}

Where marks are usually lost

• Multiplying the powers instead of adding them in aᵐ × aⁿ, or adding coefficients that should be multiplied.
• Turning a negative index into a negative number.
• Using a fractional power without taking the root first, which makes the numbers large and error-prone.
• Giving a rounded decimal when the question asks for an exact value, or rounding part-way through a calculation.
• Not simplifying surds fully (leaving √72 instead of 6√2), or not rationalising when told to.
• Stopping at a result like 12.8 × 10³ and leaving it as an answer in standard form.
• Adding or subtracting standard-form numbers without matching the powers of 10.
• Copying a calculator display such as 2.4E-5 as the final answer instead of writing 2.4 × 10⁻⁵.
• In Venn diagrams, filling the single-set regions before the overlap, so the totals do not add up.
• Using A ∪ B when the question says “and” (that is A ∩ B), or the reverse.

Official syllabus

Source: OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1, for exams May/June 2018 onwards, published by OxfordAQA. Section 3.1.1, statements N6 to N9.