Finding Approximations of Square Roots

Approximating square and cube roots of large numbers is a common task. One method is to factor out powers of 100 (for square roots) or 1000 (for cube roots) from under the radical to simplify the problem.

For example, $\sqrt{1234567} \approx \sqrt{1230000} = \sqrt{123 \times 100 \times 100} = 100\sqrt{123} \approx 100 \times 11 = 1100$.

For finding exact cube roots, like $\sqrt[3]{830584}$:

1. Group digits in threes from the right (830, 584). This indicates a two-digit answer.
2. The last digit of the number is 4. Since $4^3 = 64$, the units digit of the root is 4.
3. The first group is 830. Since $9^3 = 729 < 830$ and $10^3 = 1000 > 830$, the tens digit is 9.

The answer is 94.