Rationalise the denominator of $\frac{1}{\sqrt{11}}$
Give your answer in its simplest form.
To rationalize the denominator of the expression \(\frac{1}{\sqrt{11}}\), we aim to transform the fraction so that the denominator is a rational number. Currently, the denominator is \(\sqrt{11}\), which is an irrational number because 11 is not a perfect square. The fundamental property of square roots that we will use is that for any positive real number \(x\), the product \(\sqrt{x} \times \sqrt{x} = x\). By multiplying both the numerator and the denominator by \(\sqrt{11}\), we can effectively convert the denominator into a rational number without changing the value of the fraction.
To rationalize the expression \(\frac{1}{\sqrt{11}}\), we multiply both the numerator and the denominator by \(\sqrt{11}\):
\[\frac{1}{\sqrt{11}} = \frac{1 \times \sqrt{11}}{\sqrt{11} \times \sqrt{11}}\]
In the denominator, we have \(\sqrt{11} \times \sqrt{11} = 11\). In the numerator, we have \(1 \times \sqrt{11} = \sqrt{11}\). Therefore, the expression becomes:
\[\frac{\sqrt{11}}{11}\]
This form satisfies the condition of having a rational number in the denominator.
The expression \(\frac{\sqrt{11}}{11}\) is the simplest form of the rationalized fraction. There are no common factors between the numerator and the denominator, and the denominator is now a rational integer. Thus, we have successfully rationalized the denominator. The final result is \(\boxed{\frac{\sqrt{11}}{11}}\).
Rationalizing the denominator is a standard technique in mathematics and physics, especially when dealing with expressions involving square roots. For instance, in physics, when calculating the magnitude of a vector or working with impedance in AC circuits, you often encounter radicals in the denominator. Rationalizing these expressions makes them much easier to compare, add, and subtract. It also simplifies calculations in engineering and architecture, where precision is key when working with geometric properties like the height of equilateral triangles, where the side length involves factors of \(\sqrt{3}/2\).