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

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

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

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

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

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

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

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

= $-3+2\sqrt {7}$