what is the following quotient?\n\\( \\frac { 9 + \\sqrt { 2 } } { 4 - \\sqrt { 7 } } \\)\n\\( \\frac { 9…

Explanation:

Step1: Rationalize the denominator

Multiply the numerator and denominator by the conjugate of the denominator (4 + \sqrt{7}). [ \begin{align*} \frac{9+\sqrt{2}}{4 - \sqrt{7}}\times\frac{4+\sqrt{7}}{4+\sqrt{7}}&=\frac{(9+\sqrt{2})(4+\sqrt{7})}{(4 - \sqrt{7})(4+\sqrt{7})}\ \end{align*} ]

Step2: Expand the numerator and denominator

• Expand the numerator ((9+\sqrt{2})(4+\sqrt{7})) using the FOIL method: [ \begin{align*} (9+\sqrt{2})(4+\sqrt{7})&=9\times4+9\times\sqrt{7}+\sqrt{2}\times4+\sqrt{2}\times\sqrt{7}\ &=36 + 9\sqrt{7}+4\sqrt{2}+\sqrt{14} \end{align*} ]
• Expand the denominator ((4 - \sqrt{7})(4+\sqrt{7})) using the difference - of - squares formula ((a - b)(a + b)=a^{2}-b^{2}), where (a = 4) and (b=\sqrt{7}): [ \begin{align*} (4 - \sqrt{7})(4+\sqrt{7})&=4^{2}-(\sqrt{7})^{2}\ &=16-7\ &=9 \end{align*} ]

Answer:

(\frac{36 + 9\sqrt{7}+4\sqrt{2}+\sqrt{14}}{9}) (corresponding to the third option)