1D - Rationalising denominators

A single surd denominator

A rectangle has area $6$ and one side $\sqrt{3}$. How can its other side, $\frac{6}{\sqrt{3}}$, be written without a root in the denominator?

You already know that $\sqrt{3}\times\sqrt{3}=3$. Multiplying the denominator by $\sqrt{3}$ would remove its root, but multiplying only the denominator would change the length. Multiply the numerator by the same number to keep the fraction equal.

The key is multiplying by $1$, written as a suitable fraction. For $b>0$, the fraction $\frac{\sqrt{b}}{\sqrt{b}}$ is exactly $1$, and its numerator and denominator are nonzero.

The rectangle gives a direct example of why the new form is equivalent.

Now put a coefficient in front of the denominator's root. It stays outside the root and multiplies the new rational denominator.

Both routes depend on leaving the fraction's value unchanged.

Try the same method with a fraction that needs a final cancellation.

Two-term denominators

For a denominator such as $3-\sqrt{5}$, multiplying by $\sqrt{5}$ leaves another surd term, so we need a different multiplying fraction.

Change the subtraction to addition. When you multiply each term in the first bracket by each term in the second, the middle terms cancel:

The opposite signs make the cancellation happen. They leave a difference of squares.

You can see the same cancellation without choosing numbers.

For two different roots, the same identity gives $(\sqrt{7}+\sqrt{2})(\sqrt{7}-\sqrt{2})=7-2=5$. Choose the factor that cancels the middle terms, then simplify the whole fraction.

Keep the denominator, but change the numerator to a sum. The new feature is distributing across both numerator brackets as well.

A plus sign in the original denominator uses a minus sign in its conjugate; the difference-of-squares calculation still works.

The next denominator has a plus sign, so choose its conjugate accordingly.

Equivalent exact expressions

When an equation contains surd fractions, look at the requested answer before deciding whether to rationalise or multiply through by a denominator.

If the task asks for a rational denominator, a conjugate is a direct choice. If it asks you to find a numerator, multiplying through by the existing denominator can avoid creating another fraction. These are two applications of the same rule: perform equal operations on both sides, using nonzero denominators.

We will also compare rational and surd parts. Why is this legitimate? If rational numbers satisfy $p+q\sqrt{2}=r+s\sqrt{2}$ and $q\ne s$, rearranging would give

a rational number. But $\sqrt{2}$ is irrational. Therefore $q=s$, and then $p=r$. The same reasoning works with any irrational square root. The coefficients must be rational: allowing arbitrary real coefficients would remove this justification.

First use rationalising to identify a rational part and a surd coefficient.

Now move the unknown coefficients into the numerator. Multiplying through is shorter than rationalising that unknown numerator and forming two simultaneous equations.

For an equality of two fractions, the same approach is called cross-multiplication: if $B\ne0$ and $D\ne0$, then $\frac{A}{B}=\frac{C}{D}$ is equivalent to $AD=BC$. It is often easier to compare exact expressions after clearing both denominators.

The opening rectangle's other side is therefore exactly $2\sqrt{3}$. Rationalising makes its equality to $\frac{6}{\sqrt{3}}$ visible; neither form changes the length.

Quick Recap

• Multiply the numerator and denominator by the same nonzero expression.
• For a single root, multiply by that root; for two terms, consider the conjugate.
• Expand both numerator terms when necessary and simplify the final fraction.
• In an equation, clearing denominators can be shorter than rationalising; compare coefficients only when their stated number types justify it.

The rectangle's side is $\frac{6}{\sqrt{3}}=2\sqrt{3}$, with exactly the same value in either form.