与えられた$\sqrt{2} = 1.414$と$\sqrt{20} = 4.472$の値を用いて、以下の値を求めます。 (1) $\sqrt{18}$ (2) $\frac{2}{5\sqrt{2}}$ (3) $\sqrt{200}$ (4) $\sqrt{2000}$ (5) $\sqrt{0.02}$ (6) $\sqrt{0.2}$

1. 問題の内容

与えられた$\sqrt{2} = 1.414$と$\sqrt{20} = 4.472$の値を用いて、以下の値を求めます。

(1) $\sqrt{18}$

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

(3) $\sqrt{200}$

(4) $\sqrt{2000}$

(5) $\sqrt{0.02}$

(6) $\sqrt{0.2}$

2. 解き方の手順

(1) $\sqrt{18}$の場合：

$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$

$\sqrt{2} = 1.414$を代入すると、$3 \times 1.414 = 4.242$

(2) $\frac{2}{5\sqrt{2}}$の場合：

分母の有理化を行います。

$\frac{2}{5\sqrt{2}} = \frac{2 \times \sqrt{2}}{5\sqrt{2} \times \sqrt{2}} = \frac{2\sqrt{2}}{5 \times 2} = \frac{\sqrt{2}}{5}$

$\sqrt{2} = 1.414$を代入すると、$\frac{1.414}{5} = 0.2828$

(3) $\sqrt{200}$の場合：

$\sqrt{200} = \sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2} = 10\sqrt{2}$

$\sqrt{2} = 1.414$を代入すると、$10 \times 1.414 = 14.14$

(4) $\sqrt{2000}$の場合：

$\sqrt{2000} = \sqrt{100 \times 20} = \sqrt{100} \times \sqrt{20} = 10\sqrt{20}$

$\sqrt{20} = 4.472$を代入すると、$10 \times 4.472 = 44.72$

(5) $\sqrt{0.02}$の場合：

$\sqrt{0.02} = \sqrt{\frac{2}{100}} = \frac{\sqrt{2}}{\sqrt{100}} = \frac{\sqrt{2}}{10}$

$\sqrt{2} = 1.414$を代入すると、$\frac{1.414}{10} = 0.1414$

(6) $\sqrt{0.2}$の場合：

$\sqrt{0.2} = \sqrt{\frac{2}{10}} = \sqrt{\frac{20}{100}} = \frac{\sqrt{20}}{\sqrt{100}} = \frac{\sqrt{20}}{10}$

$\sqrt{20} = 4.472$を代入すると、$\frac{4.472}{10} = 0.4472$

3. 最終的な答え

(1) $\sqrt{18} = 4.242$

(2) $\frac{2}{5\sqrt{2}} = 0.2828$

(3) $\sqrt{200} = 14.14$

(4) $\sqrt{2000} = 44.72$

(5) $\sqrt{0.02} = 0.1414$

(6) $\sqrt{0.2} = 0.4472$