Order of operations and inverse operations
Operations are carried out in a fixed order, and the whole of algebra depends on it: brackets first, then powers and roots, then multiplication and division, then addition and subtraction. Multiplication and division rank equally and are worked left to right, and so do addition and subtraction.
The part students forget is that a fraction bar acts as a bracket. In (7 + 5) ÷ (2 + 1) both the top and the bottom are worked out before dividing, even though no brackets are printed when it is written as a stacked fraction.
Inverse operations undo each other: adding and subtracting, multiplying and dividing, squaring and square-rooting, cubing and cube-rooting. They are what lets you check an answer — if 456 ÷ 8 = 57, then 57 × 8 should return 456 — and they are the mechanism behind rearranging every equation in section 2.
Worked example
Work out 3 + 2 × (8 − 5)² ÷ 6
- Brackets first: 8 − 5 = 3, giving 3 + 2 × 3² ÷ 6
- Powers next: 3² = 9, giving 3 + 2 × 9 ÷ 6
- Multiplication and division, left to right: 2 × 9 = 18, giving 3 + 18 ÷ 6
- Still left to right: 18 ÷ 6 = 3, giving 3 + 3
- Addition last: 3 + 3 = 6
Answer: 6. Working the addition first would give 5 × 9 ÷ 6 = 7.5, which is why the order matters rather than being a convention to memorise.
- Order of operations
- the fixed order in which operations are carried out: brackets, then powers and roots, then multiplication and division, then addition and subtraction, working left to right within each rank
Factors, multiples and prime factorisation
- Factor (divisor)
- A whole number that divides into another exactly, leaving no remainder.
- Multiple
- The result of multiplying a number by a whole number — the numbers in its times table.
- Prime number
- A number with exactly two factors, itself and 1. So 1 is not prime, and 2 is the only even prime.
- Highest common factor (HCF)
- The largest number that is a factor of two or more numbers.
- Lowest common multiple (LCM)
- The smallest number that is a multiple of two or more numbers.
The unique factorisation theorem says every whole number above 1 can be written as a product of primes in exactly one way, apart from the order. That is what makes prime factorisation reliable enough to build the HCF and LCM methods on.
Find the prime factors with a factor tree: split the number into any two factors, keep splitting until every branch ends in a prime, then collect them. Write the answer in product notation using index form — 360 = 2³ × 3² × 5.
Common mistake
HCF takes the lower power, LCM takes the higher. A quick sanity check: the HCF cannot be bigger than the smaller number, and the LCM cannot be smaller than the larger one.
Systematic listing means going through possibilities in an order that guarantees none is missed — smallest first, or fixing one item and varying the rest — rather than writing them down as they occur to you.
The product rule for counting formalises it: if there are m ways of doing one task, and for each of those n ways of doing a second, then there are m × n ways of doing both. Three shirts and four ties give 3 × 4 = 12 outfits. Extend it by multiplying again for each further choice.
Powers, roots and indices
Powers of 2, 3, 4 and 5 are worth knowing on sight, because recognising 64 as 2⁶ or 4³ turns a hard question into an easy one: 2, 4, 8, 16, 32, 64, 128, 256; 3, 9, 27, 81, 243; 4, 16, 64, 256; 5, 25, 125, 625.
Estimating a root is a Higher-tier requirement and is done by trapping it between two known values. √50 lies between √49 = 7 and √64 = 8, and much closer to 7, so about 7.1.
Worked example
Simplify 16^(3/4) without a calculator.
- A fractional power means root then power: the denominator 4 is the root, the numerator 3 is the power.
- Do the root first, because it keeps the numbers small.
- The fourth root of 16 is 2, since 2 × 2 × 2 × 2 = 16.
- Now raise that to the power 3: 2³ = 8.
Answer: 8. Taking the power first would mean finding the fourth root of 4096, which is the same answer by a much harder route.
SurdsHigher tier
A surd is a root that cannot be written exactly as a fraction, so √2 and √12 are surds while √9 is not. Leaving an answer in surd form keeps it exact, which is what "give an exact answer" means.
Simplifying a surd means pulling out the largest square factor:
√12 = √(4 × 3) = √4 × √3 = 2√3
Look for the largest square factor, not just any factor. Taking √12 as √(2 × 6) gets you nowhere, because neither 2 nor 6 is a square.
The rules that make surds workable:
• √a × √b = √(ab), so √3 × √12 = √36 = 6 • √a ÷ √b = √(a/b) • a√c + b√c = (a + b)√c — you can only add or subtract surds with the same root, exactly like collecting like terms • √a × √a = a
Rationalising the denominator means removing a surd from the bottom of a fraction. Multiply top and bottom by that surd — which is multiplying by 1, so the value does not change:
6 ⁄ √3 = (6 × √3) ⁄ (√3 × √3) = 6√3 ⁄ 3 = 2√3
Common mistake
√a + √b is not √(a + b). Check it: √9 + √16 = 3 + 4 = 7, but √25 = 5. The multiplication rule works for surds; the addition rule does not exist.
- Surd
- a root that cannot be written exactly as a fraction, such as √2 or √12; leaving an answer in surd form keeps it exact
- Rationalising the denominator
- removing a surd from the bottom of a fraction by multiplying top and bottom by a suitable surd
Exact answers and standard form
Calculating exactly means leaving the answer as a fraction, a surd or a multiple of π rather than as a rounded decimal. A circle of radius 5 has area 25π cm² exactly; writing 78.5 cm² is a rounded approximation. When a question says "give your answer in terms of π", the π stays in the answer.
Standard form writes a number as A × 10ⁿ where A is at least 1 and less than 10, and n is an integer. A positive n means a large number, a negative n a small one.
To multiply or divide, deal with the numbers and the powers separately — multiply the numbers and add the powers, or divide the numbers and subtract the powers. To add or subtract, the powers must match first, or convert to ordinary numbers, combine, and convert back.
Worked example
Work out (3.6 × 10⁵) ÷ (9 × 10⁻²), giving your answer in standard form.
- Divide the numbers: 3.6 ÷ 9 = 0.4
- Subtract the powers: 5 − (−2) = 5 + 2 = 7
- That gives 0.4 × 10⁷, which is not yet in standard form because 0.4 is less than 1.
- Write 0.4 as 4 × 10⁻¹, so the answer is 4 × 10⁻¹ × 10⁷.
- Add those powers: −1 + 7 = 6.
Answer: 4 × 10⁶. The final check — is A between 1 and 10? — is where most marks in this topic are lost.
- Standard form
- a number written as a value between 1 and 10 multiplied by a power of 10
- Exact answer
- an answer left as a fraction, a surd or a multiple of π rather than as a rounded decimal
Before you practise
- Order of operations: brackets, powers and roots, multiply and divide, add and subtract — the last two pairs left to right.
- A fraction bar acts as a bracket on both the top and the bottom.
- A prime has exactly two factors; 1 is not prime.
- Write prime factorisations in index form, such as 360 = 2³ × 3² × 5.
- HCF takes common primes at the lower power; LCM takes all primes at the higher power.
- Check with HCF × LCM = the product of the two numbers.
- Product rule for counting: m ways then n ways gives m × n.
- aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1.
- A negative index means the reciprocal; a fractional index means a root — do the root first.
- Simplify a surd by taking out the largest square factor, and rationalise by multiplying top and bottom by the surd.
- √a + √b is not √(a + b).
- Standard form is A × 10ⁿ with 1 ≤ A < 10 — always check A at the end.