Q. 279 - Intermediate Algebra

Question

In the following exercises, simplify.

$\frac{\sqrt{r}+\sqrt{5}}{\sqrt{r}-\sqrt{5}}$

Step-by-Step Solution

Answer

The answer is $\frac{{\left(\sqrt{r}+\sqrt{5}\right)}^2}{r-5}$

Step 1. Given

The expression is $\frac{\sqrt{r}+\sqrt{5}}{\sqrt{r}-\sqrt{5}}$

Step 2. Rationalize the denominators of the given expressions.

$$\begin{gathered} \frac{\sqrt{r}+\sqrt{5}}{\sqrt{r}-\sqrt{5}} \\ =\frac{\sqrt{r}+\sqrt{5}}{\sqrt{r}-\sqrt{5}}\times \frac{\sqrt{r}+\sqrt{5}}{\sqrt{r}+\sqrt{5}} \\ =\frac{{\left(\sqrt{r}+\sqrt{5}\right)}^2}{{\left(\sqrt{r}\right)}^2-{\left(\sqrt{5}\right)}^2} \\ =\frac{{\left(\sqrt{r}+\sqrt{5}\right)}^2}{r-5} \end{gathered}$$