[tex]\frac{5}{3-\sqrt[]{7}}[/tex]
To simplify this fraction multiply up and down by the conjugate of the denominator
The conjugate to the denominator has the same terms but different middle sign
Then the conjugate of
[tex]3-\sqrt[]{7}\text{ is 3+}\sqrt[]{7}[/tex]
So multiply up and down by 3 + root 7
[tex]\frac{5}{3-\sqrt[]{7}}\times\frac{3+\sqrt[]{7}}{3+\sqrt[]{7}}[/tex]
Let us multiply the two denominators
[tex](3-\sqrt[]{7})(3+\sqrt[]{7})=(3)^2-(\sqrt[]{7})^2=9-7=2[/tex]
Now let us multiply the 2 numerators
[tex]5\times(3+\sqrt[]{7})=5(3)+5(\sqrt[]{7})=15+5\sqrt[]{7}[/tex]
The simplest form of the fraction is
[tex]\frac{15+5\sqrt[]{7}}{2}[/tex]
