# Difference between revisions of "Root (operation)"

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==Definition== | ==Definition== | ||

For any (not necessarily real) numbers <math>x,y,n</math>, <math>y=\sqrt[n]{x}</math> if <math>y^n=x</math>. Note that we generally take only the positive value of <math>y</math>, if we wish to take both the positive and negative roots, we write <math>\pm\sqrt[n]{x}</math>. | For any (not necessarily real) numbers <math>x,y,n</math>, <math>y=\sqrt[n]{x}</math> if <math>y^n=x</math>. Note that we generally take only the positive value of <math>y</math>, if we wish to take both the positive and negative roots, we write <math>\pm\sqrt[n]{x}</math>. | ||

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+ | ==How to compute all the roots of <math>\sqrt[n]{x}</math>== | ||

+ | |||

+ | |||

+ | A known method to compute all the roots of <math>\sqrt[n]{x}</math> is by the DeMoivre's formula. | ||

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+ | <math>\sqrt[n]{x}=\sqrt[n]{|x|}\left(\cos\frac{\theta+2\pi k}{n}+i\sin\frac{\theta+2\pi k}{n}\right)</math> , where <math>k=0,1,2,...,n-1</math> and <math>x\in\mathbb{C}</math> | ||

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+ | See that in <math>\sqrt[n]{|x|}</math> we compute its principal root. | ||

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+ | ===Example with a real number=== | ||

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+ | Compute all the roots of <math>z^4-16=0</math>. | ||

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+ | |||

+ | :First, we need to rearrange the equation . | ||

+ | |||

+ | :<math>z=\sqrt[4]{16}</math> | ||

+ | |||

+ | :See that here the "<math>x</math>" would be the number 16. | ||

+ | |||

+ | :Then, we compute <math>|16|=16</math>. | ||

+ | |||

+ | :As 2 is a pure real number, we know that <math>\theta=0</math>. | ||

+ | |||

+ | :As <math>n=4</math>, thus <math>k=0,1,2,3</math> | ||

+ | |||

+ | :We separately compute the cases <math>k=0,1,2,3</math>. | ||

+ | |||

+ | :<math>\sqrt[4]{16}=\sqrt[4]{16}\left(\cos\frac{0+2\pi\cdot0}{4}+i\sin\frac{0+2\pi\cdot0}{4}\right)\implies\boxed{2}</math> | ||

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+ | :<math>\sqrt[4]{16}=\sqrt[4]{16}\left(\cos\frac{0+2\pi\cdot1}{4}+i\sin\frac{0+2\pi\cdot1}{4}\right)\implies\boxed{2i}</math> | ||

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+ | :<math>\sqrt[4]{16}=\sqrt[4]{16}\left(\cos\frac{0+2\pi\cdot2}{4}+i\sin\frac{0+2\pi\cdot2}{4}\right)\implies\boxed{-2}</math> | ||

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+ | :<math>\sqrt[4]{16}=\sqrt[4]{16}\left(\cos\frac{0+2\pi\cdot3}{4}+i\sin\frac{0+2\pi\cdot3}{4}\right)\implies\boxed{-2i}</math> | ||

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+ | |||

+ | ==How to approximate a root== | ||

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

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+ | |||

+ | *<math>\sqrt{x}\approx\frac{x+x_0}{2\sqrt{x_0}}</math>, where <math>x_0</math> is the nearest perfect square. | ||

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+ | :Computing the square root of 5, | ||

+ | |||

+ | :We know that the nearest perfect square is 4, so, | ||

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+ | :<math>\sqrt{5}\approx\frac{5+4}{2\sqrt{4}}</math> | ||

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+ | :<math>\sqrt{5}\approx2.25</math> | ||

+ | |||

+ | :With this method you can get a little good approximation. <math>\sqrt{5}\simeq2.2360679</math> | ||

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+ | |||

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

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+ | :<math>x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}</math> , where <math>x_n</math> is a number close to the root. | ||

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

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+ | :Computing the square root of 2, | ||

+ | |||

+ | :See that it would be: | ||

+ | |||

+ | ::<math>x=\sqrt{2}</math> | ||

+ | |||

+ | ::<math>x^2-2=0</math> | ||

+ | |||

+ | ::<math>\boxed{f(x)=x^2-2}\implies f'(x)=2x</math> | ||

+ | |||

+ | |||

+ | :<math>x_{n+1}=1.4-\frac{1.4^2-2}{2\cdot1.4}</math> | ||

+ | |||

+ | :<math>x_{n+1}=1.4142857142857144</math> | ||

+ | |||

+ | :This it's very close to <math>\sqrt{2}</math>. 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== | ==See Also== | ||

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*[[Square root]], a special form of a root. | *[[Square root]], a special form of a root. | ||

− | [[Category:Operation]] | + | [[Category:Operation]] [[Category:Mathematics]] [[Category:Algebra]] |

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+ | {{stub}} |

## Revision as of 13:32, 12 January 2018

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.

*This article is a stub. Help us out by expanding it.*