33103中３・平方根・計算問題・いろいろな計算3

$(1)\;$答$8-3\sqrt{2\,} -2\sqrt{3\,}$

$\begin{eqnarray}\;\qquad \sqrt{6\,}(\,\sqrt{2\,}-\sqrt{3\,}\,)+(\,\sqrt{6\,}-\sqrt{2\,}\,)^{2}\end{eqnarray}\;\;$

$\begin{eqnarray}\quad&=& \sqrt{6\,} \times \sqrt{2\,} + \sqrt{6\,} \times (\,-\sqrt{3\,}\,)+\{\,(\,\sqrt{6\,}\,)^{2} -2 \times \sqrt{2\,} \times \sqrt{6\,} + (\,\sqrt{2\,}\,)^{2} \,\}\\[6pt]&=& 2\sqrt{3\,} -3\sqrt{2\,}+(\, 6 -4\sqrt{3\,} +2\,) \\[6pt]&=&8-3\sqrt{2\,} -2\sqrt{3\,}\end{eqnarray}\;\;$

$(2)\;$答$51-12\sqrt{5\,}$

$\begin{eqnarray}\;\qquad (\,2\sqrt{5\,}-3\,)^{2}-(\,\sqrt{3\,}+5\,)(\,\sqrt{3\,}-5\,)\end{eqnarray}\;\;$

$\begin{eqnarray} \quad &=&(\,2\sqrt{5\,}\,)^{2}-2 \times 3 \times 2\sqrt{5\,} + 3^{2} - \{\,(\,\sqrt{3\,}\,)^{2}-5^{2}\,\}\\[6pt]&=&20-12\sqrt{5\,}+9-(\,3-25\,)\\[6pt]&=&51-12\sqrt{5\,}\end{eqnarray}\;\;$

$(3)\;$答$37 + \sqrt{15\,}$

$\begin{eqnarray}\;\qquad (\,\sqrt{15\,}+2\,)(\,\sqrt{15\,}-5\,)+(\,\sqrt{30\,}+\sqrt{2\,}\,)^{2}\end{eqnarray}\;\;$

$\quad \begin{eqnarray}&=&(\,\sqrt{15\,}+2\,)(\,\sqrt{15\,}-5\,)+(\,\sqrt{30\,}+\sqrt{2\,}\,)^{2}\\[6pt]&=& (\,\sqrt{15\,}\,)^{2}+(\,2-5\,) \times \sqrt{15\,}+ \{\,2 \times (\,-5\,)\,\} +\{\,(\,\sqrt{30\,}\,)^{2}+2 \times \sqrt{2\,} \times \sqrt{30\,} +(\,\sqrt{2\,}\,)^{2}\,\} \\[6pt]&=& 15 -3\sqrt{15\,} -10 + (\,30 + 4\sqrt{15\,} + 2\,)\\[6pt]&=&37 + \sqrt{15\,}\end{eqnarray}\;\;$