### 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 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 ÷ 1.388 ^{7} = 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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