Express {\frac{8+\sqrt{7}}{2+\sqrt{7}}} in the form a+b\sqrt{7} where a and b are integers.

Answer:

Multiply {\frac{8+\sqrt{7}}{2+\sqrt{7}}} by {\frac{2-\sqrt{7}}{2-\sqrt{7}}}

\therefore \dfrac {\left( 8+\sqrt {7}\right) \times \left( 2-\sqrt {7}\right) }{\left( 2+\sqrt {7}\right) \times \left( 2-\sqrt {7}\right) }
 
= \dfrac {8\times \left( 2-\sqrt {7}\right) +\sqrt {7}\times \left( 2-\sqrt {7}\right) }{2^{2}-\left( \sqrt {7}\right) ^{2}}

= \dfrac {16-8\sqrt {7}+2\sqrt {7}-\left( \sqrt {7}\right) ^{2}}{4-7}

= \dfrac {16-6\sqrt {7}-7}{-3}

= \dfrac {9-6\sqrt {7}}{-3}

= \dfrac {9}{-3}-\dfrac {6\sqrt {2}}{-3}

= -3+2\sqrt {7}

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