Rationalizing the denominator

Rationalizing the denominator

What is 1/√2? If you work it out by hand, you have to divide by 1.41421…, which is awkward. But 1/√2 is the same as √2/2, and that is simply half of 1.41421…, so 0.7071…

Rewriting a fraction so that there is no surd in the denominator is called rationalizing the denominator. The answer is then easier to estimate, to compare and to add to other surds. (In the US, surds are called radicals, and an answer with a rationalized denominator counts as simplest radical form.)

Contents

The main idea: multiplying top and bottom

A fraction does not change if you multiply the numerator (top) and the denominator (bottom) by the same number. For example, 1/2 = 3/6.

We choose a number that makes the root in the denominator disappear. Two rules from the article Multiplying and dividing surds help us:

  • √a · √a = a
  • (√a + b)(√a − b) = a − b²

One root in the denominator

Multiply the top and the bottom by the same root that is in the denominator.

Worked example 1

1/√2

  1. Multiply top and bottom by √2: (1 · √2)/(√2 · √2).
  2. √2 · √2 = 2.
  3. 1/√2 = √2/2.

Worked example 2

6/√3

  1. Multiply top and bottom by √3: 6√3/(√3 · √3) = 6√3/3.
  2. Cancel 6 and 3: 2√3.

A number in front of the root and a root on top

Worked example 3

10/(3√5)

  1. Multiply top and bottom by √5 (the 3 stays): 10√5/(3 · 5) = 10√5/15.
  2. Cancel 10 and 15 by 5: 2√5/3.

Worked example 4

√3/√8

  1. Multiply top and bottom by √8: √3 · √8/8 = √24/8.
  2. √24 = 2√6, so 2√6/8.
  3. Cancel: √6/4.

💡 A quicker way: simplify the denominator first. √8 = 2√2, so you only need to multiply by √2: √3/(2√2) = √6/(2 · 2) = √6/4.

Two terms in the denominator

If the denominator is a sum or a difference, for example √5 − √2, multiplying by √5 does not help. Instead, multiply by the same two terms with the opposite sign in the middle: √5 + √2. This is called the conjugate. The denominator then becomes (a − b)(a + b) = a² − b², and the roots disappear.

Worked example 5

1/(√5 − √2)

  1. Multiply top and bottom by the conjugate (√5 + √2).
  2. Denominator: (√5 − √2)(√5 + √2) = (√5)² − (√2)² = 5 − 2 = 3.
  3. Numerator: 1 · (√5 + √2) = √5 + √2.
  4. 1/(√5 − √2) = (√5 + √2)/3.

Worked example 6

3/(√7 + 2)

  1. Multiply top and bottom by the conjugate (√7 − 2).
  2. Denominator: (√7)² − 2² = 7 − 4 = 3.
  3. Numerator: 3(√7 − 2).
  4. 3(√7 − 2)/3 = √7 − 2.

Check with a calculator: 3/(2.646 + 2) ≈ 0.646 and √7 − 2 ≈ 0.646 ✓.

Common mistakes

  • Multiplying only the denominator. 1/√2 is not 1/2. Always multiply both the numerator and the denominator.
  • Multiplying a two-term denominator by one root. 1/(√5 − √2) multiplied top and bottom by √5 gives (√5)/(5 − √10). There is still a root in the denominator.
  • Using the same sign instead of the opposite one. (√5 − √2)(√5 − √2) = 7 − 2√10, so the root has not disappeared.
  • Not cancelling the answer. 6√3/3 must be finished as 2√3.

Summary

  • Rationalizing the denominator = removing the surd from the denominator.
  • Remove a single root √a in the denominator by multiplying top and bottom by √a.
  • Multiply a two-term denominator √a − b by its conjugate √a + b (and the other way round). This uses the identity (x − y)(x + y) = x² − y².
  • Cancel at the end.

Try it yourself

  1. 4/√2
  2. 5/(2√10)
  3. 2/(√3 + 1)
  4. 4/(√6 − √2)
Answers:
  1. 4√2/2 = 2√2.
  2. 5√10/20 = √10/4.
  3. 2(√3 − 1)/(3 − 1) = √3 − 1.
  4. 4(√6 + √2)/(6 − 2) = √6 + √2.

Practice

👉 Next: Fractional indices (rational exponents)