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