Exact calculations — fractions, multiples of π, and surds
Examiners regularly demand "exact" answers. That means don't use a calculator's decimal approximation: leave fractions as fractions, leave π as π, leave √2 as √2.
Why "exact"?
A decimal like 3.14159… is rounded; an exact form like 3π is correct to infinitely many decimal places. Mark schemes give the A1 only for a fully simplified exact form.
Exact fraction arithmetic
Rule: never convert to decimals along the way.
- Add/subtract: common denominator, then add or subtract numerators.
- Multiply: numerators × numerators, denominators × denominators, simplify.
- Divide: keep, change, flip.
Worked example: 2/5 + 3/4. LCD = 20. = 8/20 + 15/20 = 23/20 = 1³⁄₂₀.
Multiples of π
When a question gives an exact area or circumference involving π, don't multiply π out. Treat π like a letter you can't combine with numbers.
Examples:
- Area of circle radius 6: A = πr² = 36π.
- Circumference radius 6: C = 2πr = 12π.
- Volume of cylinder, r = 5, h = 3: V = πr²h = 75π.
You can add/subtract multiples of π: 5π + 3π = 8π. You can multiply π by a number: 4 × 6π = 24π. You CANNOT add π to a non-π number: 5 + 3π stays as it is.
Surds — the basics
A surd is a root that doesn't simplify to a rational number, e.g. √2, √3, √5, √7, √10. Surds appear in Pythagoras answers, trigonometry, areas, and quadratic solutions.
Surd rules
√a × √b = √(ab).√a ÷ √b = √(a/b).p√a + q√a = (p+q)√a(only when the surd part matches).(√a)² = a.
⚠ √a + √b ≠ √(a+b). √4 + √9 = 2 + 3 = 5, not √13.
Simplifying surds
Find the largest perfect-square factor of the number under the root, then split.
Worked example: simplify √72.
- 72 = 36 × 2.
- √72 = √36 × √2 = 6√2.
Worked example: simplify 3√50.
- 50 = 25 × 2.
- √50 = 5√2.
- Multiply: 3√50 = 3 × 5√2 = 15√2.
Adding/subtracting surds
Simplify each surd first, then combine if the surd parts match.
Worked example: √18 + √8.
- √18 = √(9×2) = 3√2.
- √8 = √(4×2) = 2√2.
- Sum = 5√2.
Multiplying surds (FOIL-style)
Worked example: (2 + √3)(4 − √3).
- 2×4 + 2×(−√3) + √3×4 + √3×(−√3)
- 8 − 2√3 + 4√3 − 3
- = 5 + 2√3.
Rationalising the denominator
A fraction "isn't simplified" if the denominator contains a surd. Multiply numerator and denominator by the right thing.
- For 1/√a: multiply by √a/√a → √a/a.
- Example: 1/√3 = √3/3.
- For 1/(a + √b): multiply by (a − √b)/(a − √b) (the conjugate).
- Example: 1/(2 + √3) × (2 − √3)/(2 − √3) = (2 − √3)/(4 − 3) = 2 − √3.
⚠Common mistakes— Common mistakes (examiner traps)
- Converting to decimals along the way — answer is no longer "exact".
√a + √b = √(a+b)— wrong, never combine surds under a single root unless multiplying.- Failing to fully simplify
√72→ leaving as 2√18 instead of 6√2. - Adding non-matching surds —
√2 + √3stays as is. - Forgetting to rationalise the denominator before submitting the final answer.
➜Try this— Quick check
Simplify (√50 − √18) × √2.
- √50 = 5√2; √18 = 3√2. Difference = 2√2.
- Multiply by √2: 2√2 × √2 = 2 × 2 = 4.
AI-generated · claude-opus-4-7 · v3-deep-number