# How does this approximation of $\sqrt[R]{n}$ work?

Decades ago, when the slide rule was the norm, and calculators with the ability to calculate square roots were the cutting edge of technology, I was given an interesting formula. It went like this:

To estimate the $R$th root of $n$ using a simple calculator (where both $R$ and $n$ are positive numbers; fractions are allowed), do the following.

1. Enter $n$ into the calculator.
2. Press the $\sqrt{}$ button 12 times (i.e. calculate the 4096th root of $n$).
3. Subtract 1.
4. Divide by $R$.
6. Square the result (usually press the $\times =$ buttons) 12 times (i.e. calculate the 4096th power of the result).

Fascinatingly, this works pretty well.

For example, using this method on my calculator to find $\sqrt[8.5]{586.426}$ gives the answer with an error of just 0.05%.

This means that: $$\sqrt[R]{n} \approx \biggl(\frac{\sqrt[4096]{n} - 1}{R} + 1\biggr)^{4096}$$

Now, to me, the number $2^{12}$ seems arbitrary. So, my question is:

Why does this work?

I've tried muddling through equations to no avail.

EDIT: I've just spotted this question, which seems relevant.

• Love that you included the slide rule link, hahaha! Aug 21, 2017 at 12:52
• Yes, @ChaseRyanTaylor — I have fond memories of them! Not so much the old log tables. Aug 21, 2017 at 12:57
• Oh my goodness. That doesn't sound too bad if you know the change-of-base formula. Those tables remind me of the trigonometric ones we sometimes used in geometry. Aug 21, 2017 at 12:59
• Ah, yes, @ChaseRyanTaylor, I had forgotten about those trigonometric tables. So glad that we don't have to use them any more. These are foreign concepts to my children. Aug 21, 2017 at 13:20

You're correct that $2^{12}$ is arbitrary: it's good because it's large and a power of $2$ (so you can mash the square root button to get the $2^{12}$ root). Let the root we have available be the $k$th, so that in your case $k=4096$ (this just makes the calculation clearer), and the formula is then $$f_k(x) = \left( 1+\frac{x^{1/k}-1}{R} \right)^k$$ We want to show that for large $k$, $f_k(x) \approx x^{1/R}$. Since $k$ is large and $x$ is positive, we can use the exponential's power series to write $$x^{1/k} = e^{(1/k)\log{x}} = 1 + \frac{1}{k}\log{x} + \frac{1}{2k^2}(\log{x})^2 + O(k^{-3}),$$ where $O(g(k))$ means that the term decreases at the same rate as $g(k)$ as $k \to \infty$, so $$1 + \frac{x^{1/k}-1}{R} \approx 1 + \frac{1}{Rk}\log{x} + \frac{1}{2Rk^2}(\log{x})^2 + O(k^{-3})$$ At this point it's probably most convincing to take logs of everything, so $$\log{f_k(x)} = k\log{\left( 1+\frac{x^{1/k}-1}{R} \right)} = k\log{\left( 1 + \frac{1}{Rk}\log{x} + \frac{1}{2Rk^2}(\log{x})^2 + O(k^{-3}) \right)},$$ and since $k$ is large, the argument of the logarithm is close to $1$, and we can apply the power series for the logarithm to give $$\log{f_k(x)} = \frac{1}{R}\log{x} + \frac{1}{2Rk}(\log{x})^2 - \frac{1}{2R^2k}(\log{x})^2 + O(k^{-2})$$ Hence $$\log{\left( \frac{f_k(x)}{x^{1/R}} \right)} = \frac{R-1}{2R^2k}(\log{x})^2 + O(k^{-2}).$$ Exponentiating again gives the error in the approximation as approximately $$\frac{f_k(x)}{x^{1/R}} -1 \approx \frac{R-1}{2R^2k}(\log{x})^2,$$ which will obviously be very small for large $k$ and "ordinary-sized" $R$ and $x$ (or even large $R$, since the error depends on $1/R$).

• Man, that's clever! Thank you Aug 21, 2017 at 12:57

As you point out, $$2^{12}$$ just seems to be arbitrary and is used to denote a number that is sufficiently large. So we can reach the required result by just taking limit to infinity instead of the factor $$2^{12}$$. i.e. We need to prove that,$$\sqrt[n]{x} = \lim_{k \to \infty}\left(1+ \frac{x^{1/k} - 1}{n}\right)^k$$ which is not a hard nut to crack.

Here's the computation of the limit. Consider,\begin{align}y &=\lim_{k \to \infty}\left(1+ \frac{x^{1/k} - 1}{n}\right)^k\\& = \color{blue}{\lim_{k\to \infty} \left(1 + \frac{x^{1/k} - 1}{n}\right)}^{{\color{blue}{\left(\frac{n}{x^{1/k} -1}\right)}}\cdot \left(\frac{x^{1/k} - 1}{n}\right)\cdot k}\tag{1}\\& = \lim_{k\to \infty}\exp{\left(\frac{x^{1/k} - 1}{n}\right)\cdot k}\tag{2}\\& = \exp{\left(\color{blue}{ \lim_{k\to \infty}\frac{x^{1/k} - 1}{1/k}}\right)\cdot \frac1n} \\&= \exp\left(\log(x)\right) \cdot \frac1n\tag{3}\\& = \exp \left(\log(x^{1/n}) \right) \\& =\boxed{\sqrt[n]{x}}\end{align}

Explanation:

$$(1)$$ Here I made some adjustments in the exponent term to make it in the form of a standard limit.

$$(2)$$ Notice that, it is of the form $$\lim_{x\to\infty} \left(1 + \frac1x\right)^{x} = e.$$

$$(3)$$ Here we can simply use $$\lim_{x\to 0} \frac{x^a - 1}{a} = \log (a)$$.

• You first claim has a $k$ in it, which I guess should be an $n$. Jan 21 at 12:07
• Yes. A typo. Fixed it. Thanks! :) Jan 21 at 12:10
• @Sahaj BTW our rep points just differ by a 1 lol Jan 21 at 12:13
• Excellent, thank you! Jan 21 at 15:14