Dear Uncle Colin,

Which is larger, $\sqrt{11}$ or $5-\sqrt{5}$?

Calculate Larger Of Similar Expressions

Hi, CLOSE, and thanks for your message!

I can see two ways to do this without a calculator. The first way is the more brutal, but the second requires spotting The Exact Right Thing to work with.

### Slightly brutal

$\sqrt{1100} = \sqrt{33^2 + 11} \approx 33 + \frac{1}{6}$.

$\sqrt{5000} = \sqrt{17^3 + 87} \approx 17 + \frac{1}{3}\frac{87}{289}$, or slightly less than 17.1 – so $\sqrt{5}$ is about 1.71.

So $5 - \sqrt{5}$ is close to 3.29 and $\sqrt{11}$ is close to 3.32, and the latter is slightly larger.

(I’ve waved my hands at it slightly; after this, it’s probably a good idea to show that 3.3 is larger than one and smaller than the other.)

### Nicer

Let $x = 5 - \sqrt{11}$ (if this was a chess game, there would be a “!” after that) and $y = \sqrt{5}$.

Consider $x^3$, which is $125 - 75\sqrt{11} + 165 - 11\sqrt{11}$, or $290 - 86\sqrt{11}$. $y^3 = 5$, so $x^3 - y^3 = 285 - 86\sqrt{11}$.

Is this positive or negative? $285^2 = 81,225$ (which is easy enough to work out as $280\times290 + 25$); $86^2 = 7,396$ and $7,396 \times 11 = 81,356$, which is larger - so $x^3 - y^3 < 0$.

This implies that $x-y < 0$ – or rather, that $5 - \sqrt{11} < \sqrt{5}$.

That rearranges to $5 - \sqrt{5} < \sqrt{11}$, as before.

Hope that helps!

- Uncle Colin