How to estimate square roots fast
Most exam questions don't need a perfect square root — they need a close estimate fast enough to eliminate answer choices. This method gets you there in seconds, without long division.
The method
Find the two perfect squares the number sits between. Use the smaller one's root as your base, then adjust upward based on how far into the gap the number sits.
Worked example: √150
150 sits between 12² (144) and 13² (169) — a gap of 25 between those two squares, and 150 is 6 into that gap.
Estimate: 12 + (6/25) ≈ 12 + 0.24 ≈ 12.24. The real value is 12.247 — close enough for any multiple-choice question.
Worked example: √70
70 sits between 8² (64) and 9² (81) — a gap of 17, and 70 is 6 into that gap.
Estimate: 8 + (6/17) ≈ 8 + 0.35 ≈ 8.35. The real value is 8.367.
Why this is fast enough for exams
Most quant questions built around square roots are asking you to compare, round, or pick the closest option — not compute an exact decimal. This method gets you within a few hundredths of the true value almost every time, which is more precision than any MCQ actually needs.
Where it gets shaky
Near the middle of a very wide gap (e.g., between 10²=100 and 11²=121), the linear estimate is less accurate because square roots don't grow in a straight line. It's still usually close enough for elimination — just don't expect it to survive a request for two decimal places.
Build the foundation first
This method only works quickly if the perfect squares themselves are instant recall — which is exactly what Learn mode and Squares practice are for.