![\frac{5}{3-\sqrt[]{7}}](https://img.qammunity.org/2023/formulas/mathematics/college/wh1b58gs4ybr2hu7ym8qfnate2fum36i64.png)
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}](https://img.qammunity.org/2023/formulas/mathematics/college/383vulcy5ypv21k0r0he8xzbi3fbdt8bf7.png)
So multiply up and down by 3 + root 7
![\frac{5}{3-\sqrt[]{7}}*\frac{3+\sqrt[]{7}}{3+\sqrt[]{7}}](https://img.qammunity.org/2023/formulas/mathematics/college/qydawnl4md9ea05btd1cg45a4pr6cofqji.png)
Let us multiply the two denominators
![(3-\sqrt[]{7})(3+\sqrt[]{7})=(3)^2-(\sqrt[]{7})^2=9-7=2](https://img.qammunity.org/2023/formulas/mathematics/college/2cuk7rngnrey4g4aihiivwi93n8ytzzjzj.png)
Now let us multiply the 2 numerators
![5*(3+\sqrt[]{7})=5(3)+5(\sqrt[]{7})=15+5\sqrt[]{7}](https://img.qammunity.org/2023/formulas/mathematics/college/4y4pb34gh3vu9zk4wg6fcdmey6qqr5c4ow.png)
The simplest form of the fraction is
![\frac{15+5\sqrt[]{7}}{2}](https://img.qammunity.org/2023/formulas/mathematics/college/hssrxrogxbb07a9m0kw5db6lkeucdme801.png)