Give sqrt(85) to three significant figures without a calculator, and justify that the third figure is correct.

Give sqrt(85) to three significant figures without a calculator, and justify that the third figure is correct.

Approach: Anchor on the nearest perfect square and take a first order correction, then run one refinement step and confirm the rounding by squaring the two candidate answers.

9.22. Anchor on 81, so sqrt(85) is 9 * sqrt(1 + 4/81), and the binomial expansion of (1 + u) to the power one half gives 9 * (1 + 2/81) = 9 + 18/81 = 9.2222. One Newton iteration corrects it, since x maps to (x + 85/x)/2, and with x = 9.2222 that is (9.2222 + 9.21687)/2 = 9.21954. The true value is 9.219544, so Newton has doubled the number of correct digits in a single pass. Squaring both candidates confirms the rounding to three significant figures, since 9.21^2 falls 0.1759 short of 85 while 9.22^2 overshoots by only 0.0084.

Follow-up: How would you estimate sqrt(0.0085) to three significant figures, and where does the exponent handling go wrong?

Key concepts: Newton iteration, binomial expansion, significant figures.