How do you rationalize the denominator and simplify sqrt 55/sqrt10?

Apr 22, 2015

To keep the numbers smaller, try to reduce first. Both $55$ and $10$ are divisible by $5$. Use that to write:

$\frac{\sqrt{55}}{\sqrt{10}} = \frac{\sqrt{5} \sqrt{11}}{\sqrt{5} \sqrt{2}} = \frac{\sqrt{11}}{\sqrt{2}}$

Now rationalize tlhe denominator by multiplying by $1$ in the form: $\frac{\sqrt{2}}{\sqrt{2}}$

$\frac{\sqrt{55}}{\sqrt{10}} = \frac{\sqrt{11}}{\sqrt{2}} \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{22}}{2}$ and we're done.

Notice

I find it worth trying to reduce first. If you don't reduce first, then you'll still get the correct answer, and it hooks like this:

$\frac{\sqrt{55}}{\sqrt{10}} = \frac{\sqrt{55}}{\sqrt{10}} \frac{\sqrt{10}}{\sqrt{10}} = \frac{\sqrt{550}}{10}$

$\frac{\sqrt{55}}{\sqrt{10}} = \frac{\sqrt{550}}{10} = \frac{\sqrt{55 \cdot 10}}{10} = \frac{\sqrt{5 \cdot 11 \cdot 2 \cdot 5}}{10} = \frac{\sqrt{25 \cdot 22}}{10} = \frac{5 \sqrt{22}}{10} = \frac{\sqrt{22}}{2}$