What are two easiest numerical method to calculate $\sqrt[23]{123456789}$ by hand?

Question

What are two easiest numerical methods to calculate $\sqrt[23]{123456789}$ by hand?

I want to compare that two methods myself

Just name the methods

Answer 1

I used to remember a calculation trick where you take $12345679$ (without the $8$), and multiply it by multiples of $9$ to get repeated digits. That suggests that the number $123456789$ is very close to $10^{10}/81$ (as it indeed is), and therefore (since the log base $10$ of $9$ is about $0.954$) that the log of the number $123456789$ is about $10-1.908 = 8.092$.

Then divide $8.092$ by $23$: $23$ times $0.35$ is $8.05$, so you're only looking for about two thousandths on top of that: $0.352$. If you recognize that twice that is $0.704$ and the log base $10$ of $5$ is $0.699$ (with $6$ at $0.778$), then the desired number should be about a sixteenth of the way from $\sqrt{5}$ to $\sqrt{6}$, or $2.236+0.013 = 2.249$. Since the actual value is a bit more than $2.248$, we did all right.

Well, anyway, that's the way I would have done it mentally.

Answer 2

Note that $2^{23}=8388608$ and that $10^{23}=100000000000000000000000$

Divide by $8388608$ until a value less than $8388608$ is achieved. Then multiply by $100000000000000000000000$.

Repeat as long as you want until you achieve a value close to $1$.

I discovered that $123456789 = k \times 2^{23} \times \frac 1 {10^{23}}\times 2^{23}\times 2^{23}\times 2^{23}\times \frac 1 {10^{23}}...$

$... \times 2^{23}\times 2^{23}\times 2^{23}\times \frac 1 {10^{23}}\times 2^{23}\times 2^{23}\times 2^{23}\times 2^{23}$, where $k\approx 8.53$

Since $\sqrt [23] k \approx 1$,

$\sqrt [23] {123456789} \approx 2 \times \frac 1 {10} \times 2 \times 2 \times 2 \times \frac 1 {10} \times 2 \times 2 \times 2 \times \frac 1 {10} \times 2 \times 2 \times 2 \times 2=2.048$

Answer 3

Well, I don't know how one would name the following, neither if it is satisfying, but $$\sqrt[23]{123456789}=\sqrt[23]{1.23456789}\cdot 10^{8/23}\approx \sqrt[23]{\sqrt{1.5}}\left(\sqrt[23]{10}\right)^8=\sqrt[46]{1.5}\cdot \left(\sqrt[23]{10}\right)^8\approx2.2472,$$ and the original root is $\approx 2.2480$. Note that the rightmost roots have more familiar radicands.