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In mathematics, the general root, or the n^th 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 n^th 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:

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 n^th Root

Calculating n^th 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 n^th 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 n^th 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 b^n-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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