Answer:
-(1 + √21)/5
Explanation:
The given expression is
[tex]\frac{\sqrt[]{7}-\sqrt[]{3}}{\sqrt[]{7}-\sqrt[]{12}}[/tex]
To simplify we need to multiply and divide by the conjugate of the denominator, so we need to multiply and divide by (√7 + √12).
[tex]\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}[/tex]
Then, the expression is equal to:
[tex]\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}[/tex]
Therefore, the answer is:
-(1 + √21)/5
