Surds and rationalising the denominator

Simplifying

• sqrt(ab) = sqrt(a) x sqrt(b)
• sqrt(a/b) = sqrt(a) / sqrt(b)

Look for the largest square factor: sqrt(72) = sqrt(36) x sqrt(2) = 6sqrt(2). Like surds can then be collected: 6sqrt(2) + sqrt(2) = 7sqrt(2).

Rationalising a single surd

To remove sqrt(a) from the bottom, multiply top and bottom by sqrt(a):

6/sqrt(3) = 6sqrt(3)/3 = 2sqrt(3).

Rationalising a + sqrt(b)

Multiply top and bottom by the conjugate a - sqrt(b). The bottom becomes a difference of two squares: (a + sqrt(b))(a - sqrt(b)) = a² - b, which has no surd.

Worked example

Rationalise 4/(3 - sqrt(5)).

1. Multiply top and bottom by 3 + sqrt(5).
2. Bottom: 9 - 5 = 4.
3. Top: 4(3 + sqrt(5)).
4. Divide by 4.

Answer: 3 + sqrt(5)

Check yourself

1. Simplify sqrt(72).
   Show answer

   6√2. 72 = 36 x 2, so sqrt(72) = 6sqrt(2).

2. Simplify sqrt(50) + sqrt(18).
   Show answer

   8√2. sqrt(50) = 5sqrt(2) and sqrt(18) = 3sqrt(2). Together: 8sqrt(2).

3. Simplify sqrt(12) x sqrt(3).
   Show answer

   6. sqrt(12) x sqrt(3) = sqrt(36) = 6.

4. Rationalise 6/sqrt(3).
   • 6sqrt(3)
   • sqrt(3)/2
   • 18
   • 2sqrt(3)
   Show answer

   2sqrt(3). 6/sqrt(3) x sqrt(3)/sqrt(3) = 6sqrt(3)/3 = 2sqrt(3).

5. Expand and simplify (2 + sqrt(3))(2 - sqrt(3)).
   Show answer

   1. This is a difference of two squares: 4 - 3 = 1.

6. Rationalise 1/(sqrt(5) - 2).
   • sqrt(5) - 2
   • (sqrt(5) + 2)/9
   • (sqrt(5) + 2)/3
   • sqrt(5) + 2
   Show answer

   sqrt(5) + 2. Multiply top and bottom by sqrt(5) + 2. The bottom is 5 - 4 = 1, so the answer is sqrt(5) + 2.

7. (3 + sqrt(2))² = a + b sqrt(2). Find a.
   Show answer

   11. (3 + sqrt(2))² = 9 + 6sqrt(2) + 2 = 11 + 6sqrt(2), so a = 11.

8. (3 + sqrt(2))² = a + b sqrt(2). Find b.
   Show answer

   6. The middle terms are 3sqrt(2) + 3sqrt(2) = 6sqrt(2), so b = 6.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 1 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.