数学 - 累乗根・指数

$$ \begin{eqnarray} \sqrt4 &=& \sqrt{2 * 2} \\ &=& \sqrt{2^2} \\ &=& (\sqrt2)^2 \\ &=& \sqrt2 * \sqrt2 \\ &=& 2 \end{eqnarray} $$

$$ \begin{eqnarray} \frac{\sqrt6}{\sqrt3} &=& \frac{\sqrt{2*3}}{\sqrt3} \\ &=& \frac{\sqrt{2}*\sqrt{3}}{\sqrt3} \\ &=& \sqrt2 \\ \end{eqnarray} $$
$$ \sqrt{6} \fallingdotseq 2.4494 \\ \sqrt{3} \fallingdotseq 1.7320 \\ \frac{\sqrt{6}}{\sqrt{3}} \fallingdotseq \frac{2.4494}{1.7320} \fallingdotseq 1.41420323325635 \\ \sqrt2 \fallingdotseq 1.41421356 \\ $$

$$\begin{eqnarray} 2^3 &=& 2*2*2 = 8 \\ 2^2 &=& 2*2 = 4 \\ 2^1 &=& 2 \\ 2^0 &=& 1 \\ 2^{-1} &=& \frac{1}{2} \\ 2^{-2} &=& \frac{1}{2} * \frac{1}{2} = \frac{1}{4} \end{eqnarray}$$

$$\begin{eqnarray} 2^3 * 2^{-2} &=& 2*2*2*\frac{1}{2}*\frac{1}{2} = 2 \\ &=& 2^{3-2} = 2^1 = 2 \\ \end{eqnarray}$$

$$\begin{eqnarray} (2^4)^3 &=& 2^4 * 2^4 * 2^4 \\ &=& 2^{4+4+4} \\ &=& 2^{4*3} \\ &=& 2^{12} \\ &=& 2*2*2*2*2*2*2*2*2*2*2*2 \\ &=& 4096 \\ \end{eqnarray}$$

$$\begin{eqnarray} 3 \cdot 2^3 + 4 \cdot 2^3 &=& (3+4) \cdot 2^3 \\ &=& 7 \cdot 2^3 \\ &=& 7 * 8 = 56
\end{eqnarray}$$

$$\begin{eqnarray} 3^3 * 7^3 &=& (3*7)^3 \\
&=& 21*21*21 = 9261
\end{eqnarray}$$

$$\begin{eqnarray} 2^3 * \sqrt2 * \sqrt2 &=& 2^3 * 2^{\frac{1}{2}} * 2^{\frac{1}{2}} \\ &=& 2^{3+\frac{1}{2}+\frac{1}{2}} \\ &=& 2^4 \\ &=& 16 \\ \end{eqnarray}$$