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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:

na = b
bn = a

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:

  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
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

Calculating nth roots can be done using a similar method, with modifications to deal with n. It is highly advisable to simply use a calculator however. Even computing square roots entirely by hand is tedious. Estimating higher nth roots, even if using a calculator for intermediary steps, is significantly more tedious (and if a calculator is going to be used for intermediary steps, it would be far more efficient to simply use the calculator provided above). For those with an understanding of series, refer here for a more mathematical algorithm for calculating nth roots. For a far more inelegant, crass method that any curious or masochistic person can follow, continue to the following steps and example. To calculate na:

  1. Estimate a number b
  2. Divide a by bn-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
EX:  Find 815 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. Note that a calculator was indeed used for intermediary steps in this example.