10の5乗根(止め) - Red cat の数学よもやま話・新装開店

mathneko.hatenablog.com
mathneko.hatenablog.com

二項定理で上から止めを刺す.

$\begin{align} 1.59^5 &= (1.6 - 0.01)^5 \\ &= \left(\frac{2^4}{10} - \frac{1}{10^2}\right)^5 \\ &= \left(\frac{2^4}{10}\right)^5 - 5\left(\frac{2^4}{10}\right)^4 \left(\frac{1}{10^2}\right) + 10\left(\frac{2^4}{10}\right)^3 \left(\frac{1}{10^2}\right)^2 \\ &\quad - 10\left(\frac{2^4}{10}\right)^2 \left(\frac{1}{10^2}\right)^3 + 5\left(\frac{2^4}{10}\right)\left(\frac{1}{10^2}\right)^4 - \left(\frac{1}{10^2}\right)^5 \\ &= \frac{2^{20}}{10^5} - \frac{2^{16}\cdot 5}{10^6} + \frac{2^{12}}{10^6} - \frac{2^8}{10^7} + \frac{2^4\cdot 5}{10^9} - \frac{1}{10^{10}} \\ &= \frac{2^{20}}{10^5} - \frac{2^{15}}{10^5} + \frac{2^{12}}{10^6} - \frac{2^8}{10^7} + \frac{2^3}{10^8} - \frac{1}{10^{10}} \\ &= \frac{(2^{20}\cdot 10^5 + 2^{12}\cdot 10^4 + 2^3\cdot 10^2) - (2^{15}\cdot 10^5 + 2^8\cdot 10^3 + 1)}{10^{10}} \\ &= \frac{(104857600000 + 40960000 + 800) - (3276800000 + 256000 + 1)}{10^{10}} \\ &= \frac{104898560800 - 3277056001}{10^{10}} \\ &= \frac{101621504799}{10^{10}} \\ &= 10.1621504799 > 10 \end{align}$

結局 $1.58 \lt 10^\frac15 \lt 1.59$ だった.