Rationalise the denominator of $\frac{11}{\sqrt{5}}$. Give your answer in its simplest form.

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02-11-2026
Mathematics

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Rationalise the denominator of $\frac{11}{\sqrt{5}}$. Give your answer in its simplest form.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T10:42:29-05:00

Identify the goal: remove the radical $\sqrt{5}$ from the denominator.
Use the identity: multiply the fraction by $\frac{\sqrt{5}}{\sqrt{5}}$.
Calculate: $\frac{11 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{11\sqrt{5}}{5}$.
Final Result: $\boxed{\frac{11\sqrt{5}}{5}}$.

Explanation

Problem Analysis

The objective is to rationalize the denominator of the fraction $\frac{11}{\sqrt{5}}$. Rationalizing a denominator means rewriting a fraction so that the denominator contains only rational numbers. In this case, the denominator is $\sqrt{5}$, which is an irrational number. The goal is to eliminate the radical from the denominator without changing the overall value of the fraction.

Rationalization Method

To rationalize the denominator of a fraction in the form $\frac{a}{\sqrt{b}}$, we multiply both the numerator and the denominator by $\sqrt{b}$. By the property of fractions, multiplying by $\frac{\sqrt{b}}{\sqrt{b}}$ is equivalent to multiplying by $1$, so the value of the expression remains unchanged. In our case, we multiply by $\frac{\sqrt{5}}{\sqrt{5}}$:
\[\frac{11}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{11 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}}.\]

Calculation and Simplification

Now we perform the multiplication in the numerator and the denominator. The numerator becomes $11\sqrt{5}$. In the denominator, $\sqrt{5} \times \sqrt{5} = 5$. Substituting these values back into our expression, we get:
\[\frac{11\sqrt{5}}{5}.\]
We check for further simplification by seeing if $11$ and $5$ have common factors. Since both are prime numbers and $11$ is not divisible by $5$, the fraction is already in its simplest form.

Examples

In construction or engineering, you might encounter measurements that involve square roots, such as calculating the diagonal of a square plot of land. If a calculation results in a measurement like $\frac{11}{\sqrt{5}}$ meters, it is often more practical to rationalize the denominator to get $\frac{11\sqrt{5}}{5}$ meters. This makes it easier to estimate the numerical value using the approximation $\sqrt{5} \approx 2.236$, allowing for clearer communication of the required dimensions.