$$ \begin{align} \sqrt[3]{123456} &= 10\sqrt[3]{123.456} \ &= 50\sqrt[3]{123.456/125} \ &= 50\sqrt[3]{1 - 1.544/125} \ &= 50(1 - 0.01235)^{1/3} \ &= 50\left(1 - \frac{1}{3} \times 0.01235 - \binom{2}{1/3} \times 0.01235^2\right) \ &= 50\left(1 - \frac{1}{3} \times 0.01235 - \frac{1}{9} \times 0.01235^2\right) \ &= 50\left(1 - \frac{1}{3} \times 0.01235 - (\frac{1}{3} \times 0.01235)^2\right) \ &\approx 50 \times (1-0.004117-0.000016) \ &\approx 50 - 0.20665 \ &\approx 49.79335 \end{align} $$
The accurate result is:
49.79327984674047