Question

Express as a single fraction in simplest radical form with a rational denominator.

Answer 1

-(1 + √21)/5

Step-by-step explanation:

The given expression is

`\frac{\sqrt[]{7}-\sqrt[]{3}}{\sqrt[]{7}-\sqrt[]{12}}`

To simplify we need to multiply and divide by the conjugate of the denominator, so we need to multiply and divide by (√7 + √12).

`\begin{gathered} \frac{(\sqrt[]{7}-\sqrt[]{3})}{(\sqrt[]{7}-\sqrt[]{12})}\cdot\frac{(\sqrt[]{7}+\sqrt[]{12})}{(\sqrt[]{7}+\sqrt[]{12})} \\ =\frac{(\sqrt[]{7})^2+\sqrt[]{7}\sqrt[]{12}-\sqrt[]{3}\sqrt[]{7}-\sqrt[]{3}\sqrt[]{12}}{(\sqrt[]{7})^2+\sqrt[]{7}\sqrt[]{12}-\sqrt[]{12}\sqrt[]{7}-(\sqrt[]{12})^2} \\ =\frac{7+\sqrt[]{84}-\sqrt[]{21}-\sqrt[]{36}}{7+\sqrt[]{84}-\sqrt[]{84}-12} \end{gathered}`

Then, the expression is equal to:

`\begin{gathered} \frac{7+\sqrt[]{4\cdot21}-\sqrt[]{21}-6}{7-12} \\ =\frac{7+2\sqrt[]{21}-\sqrt[]{21}-6}{-5} \\ =\frac{1+\sqrt[]{21}}{-5}=-\frac{1+\sqrt[]{21}}{5} \end{gathered}`

Therefore, the answer is:

-(1 + √21)/5