Surds, rationalising, and exact answers (IB Math AA SL)

Key ideas

• √(ab) = √a × √b and √(a/b) = √a ÷ √b, but √(a + b) is not √a + √b.
• Take out the largest square factor: √72 = √36 × √2 = 6√2.
• Rationalise 1/√a by multiplying by √a/√a. Rationalise 1/(a + √b) by multiplying top and bottom by the conjugate a − √b, because (a + √b)(a − √b) = a² − b.
• 'Exact value' or 'in the form a + b√c' means leave the surd in: a decimal answer loses the final mark, even if it is correct.

Worked example

Without a calculator, simplify √48 + √27, and write 4/(3 − √5) in the form a + b√5.

1. √48 = √16 × √3 = 4√3 and √27 = √9 × √3 = 3√3, so √48 + √27 = 7√3.
2. Multiply 4/(3 − √5) by (3 + √5)/(3 + √5).
3. The denominator becomes 3² − 5 = 4, so the fraction is 4(3 + √5) ÷ 4 = 3 + √5.

Answer: 7√3, and 3 + √5 (a = 3, b = 1).

Exam-style SL AA questions

• SL AA Number and Algebra questions by topic — worked examples and practice questions for every topic in this unit

Related in Number and Algebra

• Quadratics, discriminant, and the nature of roots
• Arithmetic and geometric sequences
• Binomial expansion for positive integer n

Revise Surds, rationalising, and exact answers for the exam

• SL AA revision notes — every unit summarised, with the key results to learn
• SL AA past papers — specimen papers and timed practice papers
• Formulas for Number and Algebra — on the one-page SL AA formula sheet (free PDF), with the ones not in the IB booklet marked
• SL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

Stretch yourself: original algebra problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S87 · Problem S22 · Problem S86 · Problem S93. All 22 →