Rationalise the denominator of $\frac{2 \sqrt{5}}{\sqrt{6}}$. Give your answer in its simplest form.
The expression given is \(\frac{2 \sqrt{5}}{\sqrt{6}}\). The goal is to rationalize the denominator, meaning we need to transform the expression so that the square root in the denominator is eliminated. The denominator is \(\sqrt{6}\). To do this, we use the property that \(\sqrt{x} \cdot \sqrt{x} = x\) for any \(x > 0\).
To rationalize the denominator, we multiply both the numerator and the denominator by \(\sqrt{6}\). This keeps the value of the expression unchanged because we are essentially multiplying by 1: \[\frac{2 \sqrt{5}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{2 \sqrt{5} \cdot \sqrt{6}}{\sqrt{6} \cdot \sqrt{6}}.\]
Now, we simplify both the numerator and the denominator separately. For the numerator, we use the property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\): \[2 \sqrt{5} \cdot \sqrt{6} = 2 \sqrt{5 \times 6} = 2 \sqrt{30}.\] For the denominator, we have: \[\sqrt{6} \cdot \sqrt{6} = 6.\] Putting them together, the expression becomes \(\frac{2 \sqrt{30}}{6}\).
Finally, we simplify the fraction \(\frac{2 \sqrt{30}}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \[\frac{2 \sqrt{30} \div 2}{6 \div 2} = \frac{\sqrt{30}}{3}.\] Thus, the rationalized form of the expression is \(\frac{\sqrt{30}}{3}\). The final answer is \(\boxed{\frac{\sqrt{30}}{3}}\).
Rationalizing denominators is a fundamental technique used in many areas of mathematics and science to standardize expressions. For example, if you are working with physical constants or complex geometric measurements involving roots, standardizing the form helps in comparing values or performing further algebraic operations more easily. Just like simplifying fractions to their lowest terms, rationalizing denominators ensures that expressions are in a clear and consistent format for everyone to understand.