A method to compute roots other than sqrt.

[MathematicalPhysicist]

is there a method to compute roots other than sqrt?, like 10th root or 13th root of a number?

and, what are they?

 

[VietDao29]

is there a method to compute roots other than sqrt?, like 10th root or 13th root of a number?

and, what are they?

This one is a good candidate for Newton's method. We choose an arbitrary value x₀, then use:
$$x_{n + 1} = x_n - \frac{f(x_{n})}{f'(x_{n})}$$.
And then let n increases without bound to obtain the answer.
$$x = \lim_{n \rightarrow \infty} x_n$$.
For example:
Find $$\sqrt[3]{4}$$
Let $$x = \sqrt[3]{4} \Rightarrow x ^ 3 = 4 \Rightarrow x ^ 3 - 4 = 0$$
We then define f(x) := x³ - 4.
Say, we choose x₀ = 1, plug everything into a calculator, use the formula:
$$x_{n + 1} = x_n - \frac{x_{n} ^ 3 - 4}{3 x_{n} ^ 2}$$.
and we'll have:
x₁ = 2
x₂ = 1.6666667
x₃ = 1.5911111
x₄ = 1.5874097
x₅ = 1.5874010
x₆ = 1.5874010
...
So the value of x_n will converge quite fast to $$\sqrt[3]{4}$$, as n tends to infinity.

 

[matt grime]

Of course, the most obvious question is 'what is this method you have for computing arbitrary square roots'

 

[jim mcnamara]

FWIW - Logarithms work well for this. Especially if you're a programmer, and are happy with the inherent imprecision of floating point numbers.

If you take
result = (log(x) / n)

and then convert the result back ie.,

nth_root = exp(result)

you can generate all roots of x.

 

[AlphaNumeric]

If you're doing fractions then you can use Newton's binomial expansion

For instance, work out the fifth root of 31 by expanding $$(1+x)^{\frac{1}{5}}$$ with x = -1/32

$$(1+x)^{\frac{1}{5}} = 1 + \frac{1}{5}x + \frac{1}{5}\left(-\frac{4}{5}\right)\frac{1}{2!}x^{2} + ...$$

Put in x = -1/32 (which gives excellent convergence) to get

$$\left( \frac{31}{32} \right)^{\frac{1}{5}} = 1 + \frac{1}{5}\left(-\frac{1}{32}\right) + \frac{1}{5}\left(-\frac{4}{5}\right)\frac{1}{2!}\left(-\frac{1}{32}\right)^{2} + ... = 1 - \frac{1}{160} - \frac{1}{12800} = \frac{12719}{12800}$$

$$\frac{(31)^{\frac{1}{5}}}{2} = \frac{12719}{12800}$$

$$(31)^{\frac{1}{5}} = \frac{12719}{6400}$$

In decimal form this is 1.9873475. Raise it to the 5th power and get 31.00023361. A nice approximation for 2 minutes work.