# Root (operation)

The th **root** of a number , denoted by , is a common operation on numbers and a partial inverse to exponentiation. (The proper inverse is the logarithm)

## Contents

## Definition

For any (not necessarily real) numbers , if . Note that we generally take only the positive value of , if we wish to take both the positive and negative roots, we write .

## How to compute all the roots of

A known method to compute all the roots of is by the DeMoivre's formula.

, where and

See that in we compute its principal root.

### Example with a real number

Compute all the roots of .

- First, we need to rearrange the equation .

- See that here the "" would be the number 16.

- Then, we compute .

- As 2 is a pure real number, we know that .

- As , thus

- We separately compute the cases .

## How to approximate a root

There's many methods to approximate roots. Here are two:

- , where is the nearest perfect square.

- Computing the square root of 5,

- We know that the nearest perfect square is 4, so,

- With this method you can get a little good approximation.

- Also, you can use Newton-Raphson's method:

- , where is a number close to the root.

- The more times you apply this formula (consecutively), the better is the approximation that you can get.

- Computing the square root of 2,

- See that it would be:

- This it's very close to . Sometimes it can be a "very ugly bashing", but it's a method to get really good approximations. But, if couldn't get a good approximation at first, you can apply it a second time.

## See Also

- Algebra
- Square root, a special form of a root.

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