Rationalize the denominator of 1/sqrt(11). If possible, simplify the result by dividing numerator and denominator by their greatest common factor.

Step-by-step solution

1. 1

   Decide what to multiply by. The denominator is a single square root, so one copy of that same root is enough to clear it — no conjugate is needed here, because there is no sum or difference underneath:

   $\sqrt{11}\cdot\sqrt{11}=11.$

2. 2

   Multiply by a disguised 1. Multiplying numerator and denominator by the same non-zero quantity never changes the value:

   $\frac{1}{\sqrt{11}}=\frac{1}{\sqrt{11}}\cdot\frac{\sqrt{11}}{\sqrt{11}}.$

3. 3

   Carry out the multiplication. The numerator becomes $1\cdot\sqrt{11}=\sqrt{11}$ and the denominator becomes $11$:

   $\frac{\sqrt{11}}{11}.$

4. 4

   Check whether the fraction reduces. The instruction asks you to divide out the greatest common factor of numerator and denominator. Here the numerator is an irrational number $\sqrt{11}$ and the denominator is $11$; since $11$ is prime and $\sqrt{11}$ is not a multiple of $11$, the greatest common factor is $1$ and nothing cancels.

   A common slip is to cancel the $11$ under the root against the $11$ below — those are not the same quantity.

5. 5

   Confirm numerically. $1/\sqrt{11}=0.301511\ldots$ and $\sqrt{11}/11=3.316625/11=0.301511\ldots$, so the rationalized form is equal to the original.