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1) $\frac{6}{\sqrt{7}}=\frac{6}{\sqrt{7}}\cdot\frac{\sqrt{7}}{\sqrt{7}}=\frac{6\sqrt{7}}{7}$

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2) $\frac{2}{\sqrt{8}}=\frac{2}{\sqrt{4\cdot2}}=\frac{2}{2\sqrt{2}}=\frac{1}{\sqrt{2}}= \frac{1}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2}$

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3) $\frac{5}{\sqrt{12}-\sqrt{7}}=\frac{5(\sqrt{12}+\sqrt{7})}{(\sqrt{12}-\sqrt{7})(\sqrt{12}+\sqrt{7})}= \frac{5(\sqrt{12}+\sqrt{7})}{(\sqrt{12})^2-(\sqrt{7})^2}=\frac{5(\sqrt{12}+\sqrt{7})}{12-7}=$

$\frac{5(\sqrt{12}+\sqrt{7})}{5}=\sqrt{12}+\sqrt{7}$

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4) $\frac{15}{\sqrt{11}+\sqrt{6}}=\frac{15(\sqrt{11}-\sqrt{6})}{(\sqrt{11}+\sqrt{6})(\sqrt{11}-\sqrt{6})}=\frac{15(\sqrt{11}-\sqrt{6})}{(\sqrt{11})^2-(\sqrt{6})^2}=\frac{15(\sqrt{11}-\sqrt{6})}{11-6}=$

$\frac{15(\sqrt{11}-\sqrt{6})}{5}=3(\sqrt{11}-\sqrt{6})=3\sqrt{11}-3\sqrt{6}$

1) $\frac{6}{\sqrt{7}}=\frac{6\sqrt{7}}{(\sqrt{7})^2}=\frac{6\sqrt{7}}{7}$

2) $\frac{2}{\sqrt{8}}=\frac{2}{\sqrt{4\cdot2}}=\frac{2}{2\sqrt{2}}=\frac{2\sqrt{2}}{2(\sqrt{2})^2}=\frac{2\sqrt{2}}{2\cdot2}=\frac{2\sqrt{2}}{4}=\frac{\sqrt{2}}{2}$

3) $\frac{5}{\sqrt{12}-\sqrt{7}}=\frac{5(\sqrt{12}+\sqrt{7})}{(\sqrt{12}-\sqrt{7})(\sqrt{12}+\sqrt{7})}=\frac{5(\sqrt{12}+\sqrt{7})}{(\sqrt{12})^2-(\sqrt{7})^2}=\frac{5(\sqrt{12}+\sqrt{7})}{12-7}=\frac{5(\sqrt{12}+\sqrt{7})}{5}=\sqrt{12}+\sqrt{7}$