[tex]\begin{gathered} \text{If we try to rationalize} \\ \frac{5}{1+\sqrt[]{7}}\text{ using }\sqrt[]{7} \\ \\ \text{what happens is we still end up having a denominator with a radical} \end{gathered}[/tex][tex]\begin{gathered} \frac{5}{1+\sqrt[]{7}}\cdot\frac{\sqrt[]{7}}{\sqrt[]{7}},\text{ multiply by 1} \\ =\frac{5\sqrt[]{7}}{1(\sqrt[]{7})+\sqrt[]{7}(\sqrt[]{7})},\text{ distribute }\sqrt[]{7\text{ }}\text{ to numerators and denominator} \\ =\frac{5\sqrt[]{7}}{\sqrt[]{7}+7}\text{ final answer} \end{gathered}[/tex]
