Rationalise the denominator of $\frac{3 \sqrt{5}}{\sqrt{6}}$.
Give your answer in its simplest form.
The task is to rationalize the denominator of the expression \(\frac{3 \sqrt{5}}{\sqrt{6}}\). Rationalizing a denominator means eliminating the radical from the denominator by transforming the expression into an equivalent one where the denominator is a rational number. In this case, the denominator is \(\sqrt{6}\), which is an irrational number.
To rationalize the denominator, we multiply both the numerator and the denominator by \(\sqrt{6}\). This is valid because multiplying by \(\frac{\sqrt{6}}{\sqrt{6}}\) is equivalent to multiplying by \(1\):
\[\frac{3 \sqrt{5}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{3 \sqrt{5} \cdot \sqrt{6}}{\sqrt{6} \cdot \sqrt{6}}\]
Using the properties of radicals \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\), the numerator becomes \(3 \sqrt{5 \cdot 6} = 3 \sqrt{30}\). The denominator becomes \((\sqrt{6})^2 = 6\). So, the expression simplifies to:
\[\frac{3 \sqrt{30}}{6}\]
Finally, we reduce the fraction by dividing the numerator and the denominator by their greatest common divisor. Both \(3\) and \(6\) are divisible by \(3\):
\[\frac{3 \sqrt{30}}{6} = \frac{3 \div 3}{6 \div 3} \cdot \sqrt{30} = \frac{1}{2} \sqrt{30} = \frac{\sqrt{30}}{2}\]
Thus, the expression in its simplest rationalized form is \(\frac{\sqrt{30}}{2}\).
Rationalizing denominators is a fundamental skill in algebra and engineering. For example, when calculating the resonant frequency of a circuit or determining values in architectural geometry, expressions often involve square roots. Simplifying these expressions to have rational denominators makes them significantly easier to compute and compare without needing a calculator. For instance, if you have a measurement \(\frac{3}{\sqrt{2}}\) meters, expressing it as \(\frac{3\sqrt{2}}{2}\) meters helps you quickly estimate its value as approximately \(1.5 \times 1.414 = 2.121\) meters.