Rationalisation of Denominators: Method & Examples

Rationalising the denominator means rewriting a fraction so that no surd (irrational root) remains in the denominator. For example, 1/√2 becomes √2/2. The value does not change — we multiply the top and bottom by the same carefully chosen number — but the new form is the standard that mathematicians expect.

This matters for two practical reasons. First, before calculators existed, dividing by 1.414 was far harder than multiplying by it, so √2/2 was genuinely easier to compute by hand. Second, rationalised form makes it possible to add and compare fractions containing surds, since their denominators match. Exam boards and textbooks require it as the finished form.

Key Concepts

• Rationalising factor: the number you multiply top and bottom by. For 1/√a it is √a; for 1/(√a + √b) it is the conjugate √a − √b.
• Conjugate: change the sign between the two terms: the conjugate of √a + √b is √a − √b.
• Difference of squares: (x + y)(x − y) = x² − y², the identity that wipes out the surds when you multiply by the conjugate.
• Standard form: a fraction with a rational denominator and the surd (if any) sitting in the numerator.

Formulas

• 1/√a = √a/a.
• 1/(√a + √b) = (√a − √b)/(a − b).
• 1/(√a − √b) = (√a + √b)/(a − b).

How It Works

1. If the denominator is a single surd √a, multiply numerator and denominator by √a. The denominator becomes a.
2. If the denominator is √a + √b (or √a − √b), multiply numerator and denominator by its conjugate. The denominator becomes a − b.
3. Simplify the resulting fraction: cancel common factors and reduce any surds in the numerator to simplest form.

Worked Example

Rationalise 5/(√3 + √2).

1. The conjugate of √3 + √2 is √3 − √2. Multiply top and bottom by it: 5(√3 − √2) / ((√3 + √2)(√3 − √2)).
2. Denominator by difference of squares: (√3)² − (√2)² = 3 − 2 = 1.
3. So the result is 5(√3 − √2)/1 = 5√3 − 5√2. Check numerically: 5/(1.732 + 1.414) = 5/3.146 ≈ 1.589; and 5 x 1.732 − 5 x 1.414 = 8.660 − 7.070 = 1.590. Correct.

Common Misconceptions

• “Rationalising changes the value.” Wrong. You multiply by (conjugate ÷ conjugate), which equals 1, so the value is identical — only the form changes.
• “You can rationalise by squaring top and bottom.” Wrong. Squaring is not multiplying by 1; it changes the value completely.
• “The conjugate of √a + √b is −√a − √b.” Wrong. The conjugate flips only the sign between the terms: √a − √b.

Real-World Applications

• Trigonometry and geometry: exact values like sin 60° = √3/2 are kept rationalised for further calculation.
• Physics formulas with roots in denominators are rationalised before combining with other terms.
• Exam answers: most boards award full marks only for the rationalised standard form.

Rationalising is just multiplying by a clever form of 1: √a for a lone surd, the conjugate for a binomial. The denominator turns rational through the difference of squares, and the fraction keeps its value. Finish every surd fraction this way, and your answers will always be in proper form.