Root 2017

Answer: 44 and 45

Squaring numbers to get close to $2017$

$40^2=1600$ too small

$50^2=2500$ too big, so $\sqrt{2017}$ is between $40$ and $50$.

$45^2=2025$, very close but a bit too big

$44^2=1936$, too small, so $\sqrt{2017}$ is between $44$ and $45$.