JENECONKMaths Universe

Rationalising Denominators

Ages 14–18

Difficulty adapts to you

Rationalising Denominators

A rational denominator is a denominator that is a whole number or fraction, without a root. To rationalise 1/√a, multiply the top and bottom by √a; this makes the denominator √a×√a = a. For a denominator such as a + √b, multiply by its conjugate a − √b. The conjugate changes the sign and turns the denominator into a difference of squares.

Your goals

  • ✓You can remove a surd from the denominator while keeping the value unchanged.
  • ✓You can use rational denominator, rationalise, conjugate correctly.
  • ✓You can multiply the numerator and denominator by the same non-zero expression and simplify the denominator.

Words to know

rational denominator:
A denominator with no root sign in it.
rationalise:
Rewrite an equal fraction so its denominator is rational.
conjugate:
The expression made by changing + to −, or − to +, between two terms.

Examples

  • 11/√3 = √3/(√3×√3) = √3/3.
  • 25/√2 = 5√2/2 after multiplying top and bottom by √2.
  • 31/(2 + √3) × (2 − √3)/(2 − √3) = (2 − √3)/(4 − 3) = 2 − √3.
  • 41/(√5 − 2) is rationalised with the conjugate √5 + 2, not with √5 − 2 again.
  • 5Do not multiply only the denominator; multiplying top and bottom by the same non-zero expression keeps the fraction equal.

More than one way to solve it

Try each method, then choose the one you can explain and check most clearly.

Method 1

Single-root denominator

Best for: Rationalising 1/√a
  1. 1.Multiply top and bottom by √a.
  2. 2.Make the denominator √a×√a = a.
  3. 3.Simplify the numerator and denominator.

Example: 1/√3 = √3/3.

Method 2

Conjugate denominator

Best for: Rationalising a ± √b
  1. 1.Change the sign to form the conjugate.
  2. 2.Multiply top and bottom by it.
  3. 3.Use (a+b)(a−b) = a²−b².

Example: 1/(2+√3) becomes (2−√3)/(4−3) = 2−√3.

Check these

  • • Do not multiply only the denominator or use the original expression instead of the conjugate.
  • • Keep every number, symbol, label, and unit in its correct place.
  • • Choose the method that makes each step easiest to explain and check.

Helpful habits

  • • Read the whole question before choosing what to do.
  • • Show one clear step at a time and check each step.
  • • If you are stuck, draw a model, try smaller numbers, or revisit the example.

Now try your own

Create your own rationalising denominators example, solve it, and explain how you checked it.

Professor Wise Owl

Professor Wise Owl

Wisdom & Understanding

“A wise learner asks questions. Be curious!”