与えられた数式を計算し、簡略化します。問題は以下の6つです。 (1) $2^{\frac{1}{2}} + 4^{\frac{1}{2}} + 9^{\frac{1}{2}}$ (2) $(\sqrt{8} - \sqrt{27})^2$ (3) $\frac{1}{\sqrt{8}}$ (4) $\frac{\sqrt{16a^4b^2}}{2ab}$ (5) $\frac{2}{\sqrt{5}+\sqrt{3}}$ (6) $\frac{2}{\sqrt{5}+\sqrt{3}} - \frac{3}{\sqrt{5}-\sqrt{3}}$

1. 問題の内容

与えられた数式を計算し、簡略化します。問題は以下の6つです。

(1) $2^{\frac{1}{2}} + 4^{\frac{1}{2}} + 9^{\frac{1}{2}}$

(2) $(\sqrt{8} - \sqrt{27})^2$

(3) $\frac{1}{\sqrt{8}}$

(4) $\frac{\sqrt{16a^4b^2}}{2ab}$

(5) $\frac{2}{\sqrt{5}+\sqrt{3}}$

(6) $\frac{2}{\sqrt{5}+\sqrt{3}} - \frac{3}{\sqrt{5}-\sqrt{3}}$

2. 解き方の手順

(1)

$2^{\frac{1}{2}}$ は $\sqrt{2}$ と同じです。

$4^{\frac{1}{2}} = \sqrt{4} = 2$

$9^{\frac{1}{2}} = \sqrt{9} = 3$

したがって、$\sqrt{2} + 2 + 3 = \sqrt{2} + 5$

(2)

$(\sqrt{8} - \sqrt{27})^2 = (\sqrt{8})^2 - 2\sqrt{8}\sqrt{27} + (\sqrt{27})^2$

$= 8 - 2\sqrt{8 \cdot 27} + 27 = 35 - 2\sqrt{216} = 35 - 2\sqrt{36 \cdot 6} = 35 - 2 \cdot 6\sqrt{6} = 35 - 12\sqrt{6}$

(3)

$\frac{1}{\sqrt{8}} = \frac{1}{\sqrt{4 \cdot 2}} = \frac{1}{2\sqrt{2}} = \frac{1}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2 \cdot 2} = \frac{\sqrt{2}}{4}$

(4)

$\frac{\sqrt{16a^4b^2}}{2ab} = \frac{\sqrt{16} \sqrt{a^4} \sqrt{b^2}}{2ab} = \frac{4a^2|b|}{2ab} = \frac{2a|b|}{b}$

ここで、$b > 0$ と仮定すると、$\frac{2a|b|}{b} = \frac{2ab}{b} = 2a$

(5)

$\frac{2}{\sqrt{5}+\sqrt{3}} = \frac{2}{\sqrt{5}+\sqrt{3}} \cdot \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}} = \frac{2(\sqrt{5}-\sqrt{3})}{5-3} = \frac{2(\sqrt{5}-\sqrt{3})}{2} = \sqrt{5} - \sqrt{3}$

(6)

$\frac{2}{\sqrt{5}+\sqrt{3}} - \frac{3}{\sqrt{5}-\sqrt{3}}$

$\frac{2}{\sqrt{5}+\sqrt{3}} = \sqrt{5} - \sqrt{3}$ (上記(5)より)

$\frac{3}{\sqrt{5}-\sqrt{3}} = \frac{3}{\sqrt{5}-\sqrt{3}} \cdot \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}} = \frac{3(\sqrt{5}+\sqrt{3})}{5-3} = \frac{3(\sqrt{5}+\sqrt{3})}{2}$

$\sqrt{5} - \sqrt{3} - \frac{3(\sqrt{5}+\sqrt{3})}{2} = \frac{2(\sqrt{5}-\sqrt{3}) - 3(\sqrt{5}+\sqrt{3})}{2} = \frac{2\sqrt{5}-2\sqrt{3}-3\sqrt{5}-3\sqrt{3}}{2} = \frac{-\sqrt{5}-5\sqrt{3}}{2}$

3. 最終的な答え

(1) $\sqrt{2} + 5$

(2) $35 - 12\sqrt{6}$

(3) $\frac{\sqrt{2}}{4}$

(4) $2a$

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

(6) $\frac{-\sqrt{5}-5\sqrt{3}}{2}$