Rooted via 10
How many of the numbers shown are greater than 10?
Which of the numbers $3\sqrt{11}, 4\sqrt{7}, 5\sqrt{5}, 6\sqrt{3}, 7\sqrt{2}$ are greater than $10$?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Answer: $4\sqrt7$, $5\sqrt5$ and $6\sqrt3$ are greater than $10$
$\begin{align}\left(3\sqrt{11}\right)^2 &= 3\sqrt{11}\times3\sqrt{11}\\
&=3\times3\times\sqrt{11}\times\sqrt{11}\\
&=9\times11 = 99\lt100\end{align}$
So $3\sqrt{11}\lt10$
$\left(4\sqrt7\right)^2 = 16\times7 = 70 + 42\gt100$ so $4\sqrt7\gt10$
$\left(5\sqrt5\right)^2 = 25\times5 = 125 \gt100$ so $5\sqrt5\gt10$
$\left(6\sqrt3\right)^2 = 36\times3 = 108\gt100$ so $6\sqrt3\gt10$
$\left(7\sqrt2\right)^2 = 49\times2 = 98 \lt100$ so $7\sqrt2\lt10$