Estimating Square Roots and Cube Roots

Did you square both sides? That is the whole secret. You probably can’t picture $\sqrt{50}$, but you can work out $7.1 \times 7.1$ — and comparing squares tells you which root is bigger.

A calculator gives $\sqrt{28}$ to ten decimal places. The point of estimating isn’t to replace it: it’s to know, in a couple of seconds, whether the calculator’s answer is reasonable. That is exactly the check an engineer does before cutting a piece of steel.

You need these at your fingertips

…and the squares carry on: $121,\ 144,\ 169,\ 196,\ 225$.

Trap, judge, check

Trap the root between two whole numbers using the squares either side. Judge roughly where it sits. Check by squaring your guess.

The judging step is only rough, because square numbers spread out as they get bigger — from 25 to 36 is a gap of 11, from 64 to 81 is 17. So never skip the check.

Cube roots, including negative ones

Exactly the same method, with cube numbers instead of squares. Negative numbers are no problem for cube roots: $(-3)^3 = -27$, so $\sqrt[3]{-27} = -3$. Estimate as if it were positive, then put the minus sign back.

Roots of decimals

Learn a few easy ones — $\sqrt{0.09} = 0.3$, $\sqrt{0.25} = 0.5$, $\sqrt{0.49} = 0.7$ — and trapping works for decimals too. $\sqrt{0.03}$ is between $\sqrt{0.01} = 0.1$ and $\sqrt{0.04} = 0.2$; checking $0.17^2 = 0.0289$ gives $\sqrt{0.03} \approx 0.17$.