与えられた4つの複素数の計算問題を解きます。問題は以下の通りです。 (1) $\sqrt{-2}\sqrt{-8}$ (2) $\frac{\sqrt{-24}}{\sqrt{-6}}$ (3) $\frac{\sqrt{-25}}{\sqrt{5}}$ (4) $\frac{\sqrt{6}}{\sqrt{-2}}$

1. 問題の内容

与えられた4つの複素数の計算問題を解きます。問題は以下の通りです。

(1) $\sqrt{-2}\sqrt{-8}$

(2) $\frac{\sqrt{-24}}{\sqrt{-6}}$

(3) $\frac{\sqrt{-25}}{\sqrt{5}}$

(4) $\frac{\sqrt{6}}{\sqrt{-2}}$

2. 解き方の手順

(1) $\sqrt{-2}\sqrt{-8}$

$\sqrt{-2} = \sqrt{2}i$、$\sqrt{-8} = \sqrt{8}i = 2\sqrt{2}i$

$\sqrt{-2}\sqrt{-8} = (\sqrt{2}i)(2\sqrt{2}i) = 2(\sqrt{2})^2 i^2 = 2(2)(-1) = -4$

(2) $\frac{\sqrt{-24}}{\sqrt{-6}}$

$\frac{\sqrt{-24}}{\sqrt{-6}} = \sqrt{\frac{-24}{-6}} = \sqrt{4} = 2$

(3) $\frac{\sqrt{-25}}{\sqrt{5}}$

$\sqrt{-25} = \sqrt{25}i = 5i$

$\frac{\sqrt{-25}}{\sqrt{5}} = \frac{5i}{\sqrt{5}} = \frac{5}{\sqrt{5}}i = \sqrt{5}i$

(4) $\frac{\sqrt{6}}{\sqrt{-2}}$

$\sqrt{-2} = \sqrt{2}i$

$\frac{\sqrt{6}}{\sqrt{-2}} = \frac{\sqrt{6}}{\sqrt{2}i} = \frac{\sqrt{3}}{i} = \frac{\sqrt{3}i}{i^2} = \frac{\sqrt{3}i}{-1} = -\sqrt{3}i$

3. 最終的な答え

(1) -4

(2) 2

(3) $\sqrt{5}i$

(4) $-\sqrt{3}i$