Estimate answers to eliminate implausible choices quickly.
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Round to easy numbers first, then calculate. For \(463 \div 18\), round to \(500 \div 20 = 25\). The exact value is about 25.7, so the estimate is close.
Estimate a square root with the perfect squares on each side. For \(\sqrt{137}\): \(11^2 = 121\) and \(12^2 = 144\). 137 is 16 above 121 but only 7 below 144, so \(\sqrt{137} \approx 12\). Know the squares from \(1^2\) to \(20^2\) for this.
Common slips: rounding only one of the two numbers, rounding a number in the wrong direction, dividing by 2 instead of taking the square root, giving the perfect square instead of its root, and choosing the lower root when the number is nearer the upper square. Use estimates to remove answer choices that are far off.
Round first: 508 \(\to\) 500, 53 \(\to\) 50. Then \(500 \div 50 = 10\). Rounding only 508 (to 500) and dividing by 53 gives \(9\). Rounding 53 the wrong way (to 60) gives \(8\). Rounding 508 the wrong way (to 600) gives \(12\). (Each of these quotients is rounded to the nearest whole number.)
\(4^2 = 16\) and \(5^2 = 25\), and 24 falls between them, closer to 25. So \(\sqrt{24} \approx 5\). Rounding to the wrong whole number (\(4^2 = 16\) is farther from 24) gives \(4\). Dividing 24 by 2 instead of taking the square root gives \(12\). Giving the nearest perfect square, 25, instead of its square root gives \(25\).
\(20^2 = 400\) and \(21^2 = 441\), and 428 falls between them, closer to 441. So \(\sqrt{428} \approx 21\). Rounding to the wrong whole number (\(20^2 = 400\) is farther from 428) gives \(20\). Dividing 428 by 2 instead of taking the square root gives \(214\). Giving the nearest perfect square, 441, instead of its square root gives \(441\).