How to Rationalise a Denominator with Two Square Roots

What is the conjugate method?

To rationalise a denominator such as √a + √b, multiply numerator and denominator by √a − √b. The denominator becomes a − b because (x + y)(x − y) = x² − y². This method requires the conjugate to be nonzero; simplify special cases before multiplying.

Worked example: 1/(√5 + √2)

Multiply by (√5 − √2)/(√5 − √2). The numerator becomes √5 − √2 and the denominator becomes 5 − 2 = 3. Therefore the simplified answer is (√5 − √2)/3. We changed the appearance of the expression by multiplying by 1; its value stayed the same.

Check by multiplying the result by the original denominator: [(√5 − √2)/3](√5 + √2) = (5 − 2)/3 = 1. This reverse check catches many sign errors.

What if the denominator has a minus sign?

For 2/(3 − √5), use 3 + √5. The denominator is 9 − 5 = 4, and the numerator is 6 + 2√5. Dividing both terms by 4 gives (3 + √5)/2.

Do not write 9 + 5 in the denominator. The two middle terms cancel, leaving a difference of squares. Also multiply the entire numerator, not just one of its terms.

A special case to simplify first

For 1/(√3 + √3), the supposed conjugate is zero, so multiplying by its ratio would be undefined. Combine the terms: 1/(2√3) = √3/6. A denominator equal to zero is undefined and cannot be repaired by rationalisation.

Practice with answers

• 1/(√7 + √3) = (√7 − √3)/4.
• 3/(√6 − √3) = √6 + √3.
• 1/(2 + √3) = 2 − √3.

Check each answer by multiplying it by the original denominator. Rationalising is an exact algebraic step, so decimal approximations are unnecessary.

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