Estimating Powers and Roots [H]
Estimating Powers and Roots
Most square roots and cube roots are irrational — their decimal representations are non-terminating and non-repeating, and no exact fraction represents them. When a calculator is unavailable, the bracketing method finds an approximate value by identifying which two consecutive perfect powers the target lies between.
The method relies on monotonicity: for positive bases greater than 1, larger inputs give larger powers. So if \( k^2 < N < (k+1)^2 \), then \( k < \sqrt{N} < k+1 \).
Bracketing on the Number Line — Estimating \( \sqrt{50} \)
Since \( 7^2 = 49 \) and \( 8^2 = 64 \), we have \( 7 < \sqrt{50} < 8 \). Because 50 is only 1 above 49 but 14 below 64, \( \sqrt{50} \) is very close to 7. More precisely, \( \sqrt{50} \approx 7.07 \).
Worked Examples
Estimate \( \sqrt{80} \) to 1 decimal place.
Bracket: \( 8^2 = 64 < 80 < 81 = 9^2 \), so \( 8 < \sqrt{80} < 9 \). Very close to 9.
Try 8.9: \( 8.9^2 = 79.21 \) (too low). Try 8.95: \( 8.95^2 = 80.10 \) (just above).
Since \( 79.21 < 80 < 80.10 \), we have \( \sqrt{80} \approx 8.9 \) to 1 d.p.
Estimate \( \sqrt[3]{50} \) to the nearest integer.
\[ 3^3 = 27 quad 4^3 = 64 \]Since \( 27 < 50 < 64 \), we have \( 3 < \sqrt[3]{50} < 4 \). 50 is closer to 64 (distance 14) than to 27 (distance 23), so the root is closer to 4.
To the nearest integer: \( \sqrt[3]{50} \approx 4 \). (More precisely, \( \approx 3.68 \).)
Estimate \( \sqrt{75} \) to 1 decimal place.
Bracket: \( 8^2 = 64 < 75 < 81 = 9^2 \).
| Trial \(x\) | \(x^2\) | Verdict |
|---|---|---|
| 8.5 | 72.25 | Too low |
| 8.7 | 75.69 | Too high |
| 8.6 | 73.96 | Too low |
| 8.66 | 74.9956 | Just below 75 |
| 8.67 | 75.1689 | Just above 75 |
\( \sqrt{75} \approx 8.7 \) to 1 d.p.
Key Takeaways
- To estimate \( \sqrt{N} \): find \( k \) and \( k+1 \) such that \( k^2 < N < (k+1)^2 \). Then \( k < \sqrt{N} < k+1 \).
- To estimate \( \sqrt[3]{N} \): find \( k \) and \( k+1 \) such that \( k^3 < N < (k+1)^3 \).
- The closer \( N \) is to the larger perfect power, the closer the root is to the upper integer bound.
- Trial and improvement refines the bracket to a required number of decimal places by systematically squaring or cubing test values.
- Knowing squares to \( 15^2 \) and cubes to \( 10^3 \) makes the initial bracket step quick and reliable.