Square Roots

Squaring and square-rooting are inverse operations: since $7^2 = 49$, we have $\sqrt{49} = 7$. A perfect square is a number whose square root is a whole number — $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \dots$ Knowing these on sight is half the skill.

Most numbers aren't perfect squares, and their roots aren't whole. For those, you estimate by trapping the number between the two nearest perfect squares — that tells you which two whole numbers the root lies between. (Geometry rarely needs the exact decimal; knowing $\sqrt{50}$ is "a bit more than $7$" is usually enough.)

Perfect squares — recall the pair. $\sqrt{64} = 8$ because $8^2 = 64$.

Estimate by trapping. Where is $\sqrt{50}$? Find the perfect squares just below and above $50$:

$49 < 50 < 64 \quad\Rightarrow\quad 7 < \sqrt{50} < 8$

So $\sqrt{50}$ lies between $7$ and $8$ (and closer to $7$, since $50$ is just past $49$).

Where it pays off. A right triangle with legs $6$ and $8$ has hypotenuse $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ — squares going in, a square root coming out.

To place a non-perfect root, find the perfect square just below and just above. Predict before you check.