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

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