题目

Problem

Answer this question showing each stage of your working.

(a) Simplify

\begin{align*} \frac{1}{4-2\sqrt2}, \end{align*}

giving your answer in the form $a+b\sqrt2$ where $a$ and $b$ are rational numbers.

(2)

(b) Hence, or otherwise, solve the equation

\begin{align*} 4x=2\sqrt2x+20\sqrt2, \end{align*}

giving your answer in the form $p+q\sqrt2$ where $p$ and $q$ are rational numbers.

(3)

解答

展开

分母含有根号，用共轭式 $4+2\sqrt2$ 有理化。

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\begin{align*} \frac{1}{4-2\sqrt2} =&\,\frac{1}{4-2\sqrt2}\cdot\frac{4+2\sqrt2}{4+2\sqrt2}\\ =&\,\frac{4+2\sqrt2}{16-8}\\ =&\,\frac12+\frac14\sqrt2. \end{align*}

展开

先把含 $x$ 的项放在同一边，再用上一问的结果处理分母。

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\begin{align*} 4x=&\,2\sqrt2x+20\sqrt2\\ (4-2\sqrt2)x=&\,20\sqrt2\\ x=&\,\frac{20\sqrt2}{4-2\sqrt2}. \end{align*}

Using part (a),

\begin{align*} x =&\,20\sqrt2\left(\frac12+\frac14\sqrt2\right)\\ =&\,10\sqrt2+10\\ =&\,10+10\sqrt2. \end{align*}