Rationalising Denominators

Rationalising Denominators

Rationalising removes a surd from the denominator without changing a value. Multiply numerator and denominator by the same non-zero expression. For a single root use that root; for a binomial denominator use the conjugate, which changes the sign between terms. The product of conjugates gives a difference of squares.

Worked reasoning

$\frac{3}{\sqrt5}=\frac{3}{\sqrt5}\cdot\frac{\sqrt5}{\sqrt5}=\frac{3\sqrt5}{5}$. The factor $\sqrt5/\sqrt5$ is 1, so the fraction stays equivalent.

Exam method

1. Identify the full denominator. 2. Choose a factor that removes its radical part. 3. Multiply top and bottom, simplify, and leave no radical in the denominator.

Why the denominator matters

The convention of rationalising is not a trick for changing values; it leaves an exact answer in a standard usable form. For binomial denominators, expand both numerator and denominator carefully. The middle terms of conjugates cancel, which is why $(a+b)(a-b)=a^2-b^2$. Write a final line that shows the denominator is rational. This makes it easy for a marker to see that the requested form, not just an equivalent form, has been reached.

One-minute retrieval: Rationalising Denominators

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.