Rationalising Denominators and Surd Equations
Rationalising a denominator means rewriting a fraction such as 1/sqrt(3) so that no surd is left on the bottom, by multiplying top and bottom by the surd or by the conjugate of the denominator. A surd equation contains a square root of the unknown, solved by isolating the root, squaring both sides, then checking for extraneous solutions. Both skills are examined in AQA Further Maths papers.
Before you start
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Method
- To rationalise a denominator like a/sqrt(b), multiply the numerator and denominator by sqrt(b) so the denominator becomes the integer b.
- To rationalise a denominator with two terms, such as a/(sqrt(b) + c), multiply the numerator and denominator by the conjugate (sqrt(b) - c).
- Use the difference of two squares on the denominator so the surd cancels out, leaving a rational number.
- Expand the numerator carefully, then simplify by collecting rational and surd terms separately.
- To solve a surd equation, isolate the square root term on one side of the equation before squaring.
- Square both sides to remove the root, solve the resulting equation, then substitute each solution back into the original equation to reject any extraneous root.
Worked example
Rationalise the denominator of 6/(sqrt(7) - 2), simplifying fully.
- Multiply the numerator and denominator by the conjugate of the denominator, sqrt(7) + 2.
- The denominator becomes (sqrt(7) - 2)(sqrt(7) + 2) = 7 - 4 = 3, using the difference of two squares.
- The numerator becomes 6(sqrt(7) + 2) = 6sqrt(7) + 12.
- Write the fraction as (6sqrt(7) + 12) / 3.
- Divide every term by 3 to simplify fully.
- Final answer: 2sqrt(7) + 4.
Practice questions
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Q1Rationalise the denominator of 1/sqrt(6), simplifying fully.Show answer
Answer: sqrt(6)/6 (multiply top and bottom by sqrt(6))
Q2Rationalise the denominator of 10/sqrt(5), simplifying fully.Show answer
Answer: 2sqrt(5) (10sqrt(5)/5 simplifies to 2sqrt(5))
Q3Solve sqrt(x) = 9.Show answer
Answer: x = 81 (square both sides)
Q4Solve sqrt(x - 4) = 3.Show answer
Answer: x = 13 (square both sides: x - 4 = 9)
Q5Rationalise the denominator of 3/(sqrt(2) + 1), simplifying fully.Show answer
Answer: 3sqrt(2) - 3 (multiply by the conjugate sqrt(2) - 1)
Q6Solve sqrt(2x + 1) - 3 = 2, checking your solution satisfies the original equation.Show answer
Answer: x = 12 (sqrt(2x+1) = 5, so 2x + 1 = 25)
Exam-style questions
Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.
Rationalise the denominator of 8/sqrt(2), simplifying fully.
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Solve the equation sqrt(3x - 5) = 7, showing your working.
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Solve sqrt(5x + 6) = x, showing why any extraneous solution must be rejected.
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See real GCSE Further Maths past-paper questions, with official mark schemes →
Free printable worksheet
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