To make a division of complex numbers, we need to multiply for the conjugate of the denominator:
[tex]\frac{-\sqrt{11}-i\sqrt{7}}{-\sqrt{10}-i\sqrt{6}}[/tex]First, we can factor out a (-1) on top and bottom:
[tex]\frac{-\sqrt{11}-i\sqrt{7}}{-\sqrt{10}-i\sqrt{6}}=\frac{(-1)(\sqrt{11}+i\sqrt{7})}{(-1)(\sqrt{10}+i\sqrt{6}}[/tex]And the two (-1) can cel each other, and now we multiply by the conjugate:
[tex]\frac{\sqrt{11}+i\sqrt{7}}{\sqrt{10}+i\sqrt{6}}\cdot\frac{\sqrt{11}-i\sqrt{7}}{\sqrt{10}-i\sqrt{6}}[/tex]And solve:
[tex]\frac{(\sqrt{11}+i\sqrt{7})(\sqrt{10}-i\sqrt{6})}{(\sqrt{10}+i\sqrt{6})(\sqrt{10}-i\sqrt{6})}=\frac{\sqrt{11}\sqrt{10}^-i\sqrt{6}\sqrt{11}+i\sqrt{7}\sqrt{10}-i^2\sqrt{7}\sqrt{6}}{(\sqrt{10})^2+i\sqrt{6}\sqrt{10}-i\sqrt{6}\sqrt{10}-i^2(\sqrt{6})^2}[/tex]Two terms simplyfy in the denominator, and the square root of the denominator cancel out:
[tex]\frac{\sqrt{11\cdot10}^-\imaginaryI\sqrt{6\cdot11}+\imaginaryI\sqrt{7\cdot10}-(-1)\sqrt{7\cdot6}}{10-(-1)\cdot6}[/tex]And now we can make the multiplycations:
[tex]\frac{\sqrt{110}-i\sqrt{66}+i\sqrt{70}+\sqrt{42}}{16}[/tex]Now we can factor out i, and solve to get the number in the form a + ib
[tex]\frac{\sqrt{110}-\sqrt{42}}{16}+i\frac{\sqrt{70}-\sqrt{66}}{16}[/tex]