Estimating a Root
Some common roots include the square root, where n = 2, and the cubed root, where n = 3. Calculating square roots and nth roots is fairly intensive. It requires estimation and trial and error. There exist more precise and efficient ways to calculate square roots, but below is a method that does not require significant understanding of more complicated math concepts. To calculate √a:
- Estimate a number b
- Divide a by b. If the number c returned is precise to the desired decimal place, stop.
- Average b and c and use the result as a new guess
- Repeat step two
| EX: | Find √27 to 3 decimal places |
| Guess: 5.125 27 ÷ 5.125 = 5.268 (5.125 + 5.268)/2 = 5.197 27 ÷ 5.197 = 5.195 (5.195 + 5.197)/2 = 5.196 27 ÷ 5.196 = 5.196 |
Estimating an nth Root
Calculating nth roots can be done using a similar method, with modifications to deal with n. While computing square roots entirely by hand is tedious. Estimating higher nth roots, even if using a calculator for intermediary steps, is significantly more tedious. For those with an understanding of series, refer here for a more mathematical algorithm for calculating nth roots. For a simpler, but less efficient method, continue to the following steps and example. To calculate n√a:
- Estimate a number b
- Divide a by bn-1. If the number c returned is precise to the desired decimal place, stop.
- Average: [b × (n-1) + c] / n
- Repeat step two
| EX: | Find 8√15 to 3 decimal places |
| Guess: 1.432 15 ÷ 1.4327 = 1.405 (1.432 × 7 + 1.405)/8 = 1.388 15 ÷ 1.3887 = 1.403 (1.403 × 7 + 1.388)/8 = 1.402 |
It should then be clear that computing any further will result in a number that would round to 1.403, making 1.403 the final estimate to 3 decimal places.
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