## Rationalising Denominators

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It is often easier to work with fractions that have rational denominators instead of surd denominators. By rationalising the denominator, we convert a fraction with a surd in the denominator to a fraction that has a rational denominator.

## Example

### Question

Rationalise the denominator: \[\cfrac{5x-16}{\sqrt{x}}\]

### Multiply the fraction by \(\cfrac{\sqrt{x}}{\sqrt{x}}\)

Notice that \(\cfrac{\sqrt{x}}{\sqrt{x}}=1\), so the value of the fraction has not been changed.

\[\cfrac{5x – 16}{\sqrt{x}} \times \cfrac{\sqrt{x}}{\sqrt{x}} = \cfrac{\sqrt{x}(5x – 16)}{\sqrt{x} \times \sqrt{x}}\]

### Simplify the denominator

\begin{align*} &= \cfrac{\sqrt{x}(5x – 16)}{( \sqrt{x})^2} \\ &= \cfrac{\sqrt{x}(5x – 16)}{x} \end{align*}

The term in the denominator has changed from a surd to a rational number. Expressing the surd in the numerator is the preferred way of writing expressions.

## Example

### Question

Write the following with a rational denominator: \[\cfrac{y-25}{\sqrt{y}+5}\]

### Multiply the fraction by \(\cfrac{\sqrt{y}-5}{\sqrt{y}-5}\)

To eliminate the surd from the denominator, we must multiply the fraction by an expression that will result in a difference of two squares in the denominator.

\[\cfrac{y-25}{\sqrt{y}+5} \times \cfrac{\sqrt{y}-5}{\sqrt{y}-5}\]

### Simplify the denominator

\begin{align*} &= \cfrac{(y-25)(\sqrt{y}-5)}{(\sqrt{y}+5)(\sqrt{y}-5)} \\ &= \cfrac{(y-25)(\sqrt{y}-5)}{(\sqrt{y})^2 -25} \\ &= \cfrac{(y-25)(\sqrt{y}-5)}{y-25} \\ &= \sqrt{y}-5 \end{align*}