$\sqrt{2} = 1.414$, $\sqrt{3} = 1.732$, $\sqrt{5} = 2.236$ であるとき、以下の値を求めよ。 (1) $\sqrt{12}$ (2) $\sqrt{125}$ (3) $\sqrt{0.02}$ (4) $\sqrt{2000}$ (5) $\frac{1}{\sqrt{2}}$ (6) $\frac{3}{5\sqrt{3}}$

1. 問題の内容

$\sqrt{2} = 1.414$, $\sqrt{3} = 1.732$, $\sqrt{5} = 2.236$ であるとき、以下の値を求めよ。

(1) $\sqrt{12}$

(2) $\sqrt{125}$

(3) $\sqrt{0.02}$

(4) $\sqrt{2000}$

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

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

2. 解き方の手順

(1) $\sqrt{12}$

$\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} = 2 \times 1.732 = 3.464$

(2) $\sqrt{125}$

$\sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5} = 5 \times 2.236 = 11.18$

(3) $\sqrt{0.02}$

$\sqrt{0.02} = \sqrt{\frac{2}{100}} = \frac{\sqrt{2}}{\sqrt{100}} = \frac{\sqrt{2}}{10} = \frac{1.414}{10} = 0.1414$

(4) $\sqrt{2000}$

$\sqrt{2000} = \sqrt{1000 \times 2} = \sqrt{100 \times 10 \times 2} = \sqrt{100} \times \sqrt{10 \times 2} = 10\sqrt{20} = 10\sqrt{4 \times 5} = 10 \times 2\sqrt{5} = 20\sqrt{5} = 20 \times 2.236 = 44.72$

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

$\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} = \frac{1.414}{2} = 0.707$

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

$\frac{3}{5\sqrt{3}} = \frac{3}{5\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{5 \times 3} = \frac{\sqrt{3}}{5} = \frac{1.732}{5} = 0.3464$

3. 最終的な答え

(1) 3.464

(2) 11.18

(3) 0.1414

(4) 44.72

(5) 0.707

(6) 0.3464