Question

I don't get how this expression simplifies to this answer. I'm supposed to distribute the 5, but I have no idea where that +5 in answer comes from.

Answer 1

`\frac{5}{3-\sqrt[]{7}}`

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

`3-\sqrt[]{7}\text{ is 3+}\sqrt[]{7}`

So multiply up and down by 3 + root 7

`\frac{5}{3-\sqrt[]{7}}*\frac{3+\sqrt[]{7}}{3+\sqrt[]{7}}`

Let us multiply the two denominators

`(3-\sqrt[]{7})(3+\sqrt[]{7})=(3)^2-(\sqrt[]{7})^2=9-7=2`

Now let us multiply the 2 numerators

`5*(3+\sqrt[]{7})=5(3)+5(\sqrt[]{7})=15+5\sqrt[]{7}`

The simplest form of the fraction is

`\frac{15+5\sqrt[]{7}}{2}`