The value of $\frac{1}{\sqrt{12 - \sqrt{140}}} - \frac{1}{\sqrt{8 - \sqrt{60}}} - \frac{2}{\sqrt{10 + \sqrt{84}}}$ is

Concept & Formula To simplify a surd of the form $\sqrt{a \pm 2\sqrt{b}}$, we find two numbers whose sum is $a$ and product is $b$. The expression then becomes a perfect square $(\sqrt{x} \pm \sqrt{y})^2$. If the $2$ is missing before the inner root, extract it from inside the root by dividing the inner number by $4$. $$ (x \pm y)^2 = x^2 + y^2 \pm 2xy $$ Step-by-Step Solution * Evaluate the first denominator: $\sqrt{12 - \sqrt{140}}$ Extract a $4$ from $140$ to get a $2$ outside: $\sqrt{12 - 2\sqrt{35}}$ We need numbers that add to $12$ and multiply to $35$. They are $7$ and $5$. So, $\sqrt{12 - 2\sqrt{35}} = \sqrt{7} - \sqrt{5}$ * Evaluate the second denominator: $\sqrt{8 - \sqrt{60}}$ Extract a $4$ from $60$: $\sqrt{8 - 2\sqrt{15}}$ Numbers adding to $8$ and multiplying to $15$ are $5$ and $3$. So, $\sqrt{8 - 2\sqrt{15}} = \sqrt{5} - \sqrt{3}$ * Evaluate the third denominator: $\sqrt{10 + \sqrt{84}}$ Extract a $4$ from $84$: $\sqrt{10 + 2\sqrt{21}}$ Numbers adding to $10$ and multiplying to $21$ are $7$ and $3$. So, $\sqrt{10 + 2\sqrt{21}} = \sqrt{7} + \sqrt{3}$ * Rationalize each term in the main expression: Term 1: $\frac{1}{\sqrt{7} - \sqrt{5}} = \frac{\sqrt{7} + \sqrt{5}}{7 - 5} = \frac{\sqrt{7} + \sqrt{5}}{2}$ Term 2: $\frac{1}{\sqrt{5} - \sqrt{3}} = \frac{\sqrt{5} + \sqrt{3}}{5 - 3} = \frac{\sqrt{5} + \sqrt{3}}{2}$ Term 3: $\frac{2}{\sqrt{7} + \sqrt{3}} = \frac{2(\sqrt{7} - \sqrt{3})}{7 - 3} = \frac{2(\sqrt{7} - \sqrt{3})}{4} = \frac{\sqrt{7} - \sqrt{3}}{2}$ * Combine all rationalized terms: $$ \frac{\sqrt{7} + \sqrt{5}}{2} - \frac{\sqrt{5} + \sqrt{3}}{2} - \frac{\sqrt{7} - \sqrt{3}}{2} $$ $$ = \frac{\sqrt{7} + \sqrt{5} - \sqrt{5} - \sqrt{3} - \sqrt{7} + \sqrt{3}}{2} $$ $$ = \frac{0}{2} = 0 $$ Exam Strategy & Shortcut Instead of writing out the full rationalization step, recognize the pattern: $\frac{1}{\sqrt{a} - \sqrt{b}} = \frac{\sqrt{a} + \sqrt{b}}{a - b}$. When you identify the differences between the numbers inside the roots (e.g., $7 - 5 = 2$, $5 - 3 = 2$), you can immediately put them all over a common denominator of $2$ and cancel the alternating positive and negative terms mentally to save crucial seconds. Common Pitfall Failing to distribute the negative signs correctly when combining the fractions. Writing $-\frac{\sqrt{5} + \sqrt{3}}{2}$ and accidentally calculating it as $-\sqrt{5} + \sqrt{3}$ instead of $-\sqrt{5} - \sqrt{3}$ is a frequent algebraic mistake that leads to non-zero answers. Final Answer Therefore, the correct answer is 0.