About Roots

In mathematics, the general root, or the nth root of a number a is another number b that when multiplied by itself n times, equals a. In equation format:

n√a = b
bn = a

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 a significant understanding of more complicated math concepts. To calculate √a:

1. Estimate a number b
2. Divide a by b. If the number c returned is precise to the desired decimal place, stop.
3. Average b and c and use the result as a new guess
4. Repeat step two

Example: 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:

1. Estimate a number b
2. Divide a by b^n-1. If the number c returned is precise to the desired decimal place, stop.
3. Average: [b × (n-1) + c] / n
4. Repeat step two

Example: 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.