$\frac{1}{(\sqrt{9} - \sqrt{8})} - \frac{1}{(\sqrt{8} - \sqrt{7})} + \frac{1}{(\sqrt{7} - \sqrt{6})} - \frac{1}{(\sqrt{6} - \sqrt{5})} + \frac{1}{(\sqrt{5} - \sqrt{4})}$ is equal to

### Concept & Formula When dealing with fractions that have irrational numbers (surds) in the denominator, the standard mathematical procedure is **rationalization**. To rationalize a denominator of the form $\sqrt{a} - \sqrt{b}$, multiply both the numerator and the denominator by its conjugate, $\sqrt{a} + \sqrt{b}$. $$ \frac{1}{\sqrt{a} - \sqrt{b}} = \frac{\sqrt{a} + \sqrt{b}}{(\sqrt{a} - \sqrt{b})(\sqrt{a} + \sqrt{b})} = \frac{\sqrt{a} + \sqrt{b}}{a - b} $$ Notice that for every term in this sequence, the difference between the numbers under the square roots is exactly $1$ (e.g., $9 - 8 = 1$, $8 - 7 = 1$). This means the denominators will all simplify to $1$. ### Step-by-Step Solution * **Calculation / Deduction:** * Let's rationalize each term individually based on the formula above. * **First Term:** $$ \frac{1}{\sqrt{9} - \sqrt{8}} = \frac{\sqrt{9} + \sqrt{8}}{9 - 8} = \sqrt{9} + \sqrt{8} $$ * **Second Term:** $$ \frac{1}{\sqrt{8} - \sqrt{7}} = \frac{\sqrt{8} + \sqrt{7}}{8 - 7} = \sqrt{8} + \sqrt{7} $$ * **Third Term:** $$ \frac{1}{\sqrt{7} - \sqrt{6}} = \sqrt{7} + \sqrt{6} $$ * **Fourth Term:** $$ \frac{1}{\sqrt{6} - \sqrt{5}} = \sqrt{6} + \sqrt{5} $$ * **Fifth Term:** $$ \frac{1}{\sqrt{5} - \sqrt{4}} = \sqrt{5} + \sqrt{4} $$ * Now, substitute these simplified terms back into the original alternating sequence. Be very careful with the negative signs distributing to both parts of the rationalized terms! $$ (\sqrt{9} + \sqrt{8}) - (\sqrt{8} + \sqrt{7}) + (\sqrt{7} + \sqrt{6}) - (\sqrt{6} + \sqrt{5}) + (\sqrt{5} + \sqrt{4}) $$ * Open the brackets and distribute the signs: $$ \sqrt{9} + \sqrt{8} - \sqrt{8} - \sqrt{7} + \sqrt{7} + \sqrt{6} - \sqrt{6} - \sqrt{5} + \sqrt{5} + \sqrt{4} $$ * Notice how all the intermediate terms cancel each other out: $$ \sqrt{9} + 0 + 0 + 0 + 0 + \sqrt{4} $$ * We are left with just the first and last parts of the sequence: $$ \sqrt{9} + \sqrt{4} $$ $$ 3 + 2 = 5 $$ ### Exam Strategy & Shortcut This is a classic "telescoping series" pattern disguised with surds. As soon as you spot an alternating sequence of $\frac{1}{\sqrt{x} - \sqrt{x-1}}$, recognize that the denominators evaluate to $1$, and the middle terms will always cancel out. You only need to look at the very first part of the first term ($\sqrt{9}$) and the very last part of the last term ($\sqrt{4}$). Because the last term has a positive sign in the sequence, it becomes $\sqrt{9} + \sqrt{4} = 3 + 2 = 5$. This observation turns a 2-minute calculation into a 10-second mental math problem. ### Common Pitfall The most frequent error is forgetting to distribute the negative sign to the *entire* rationalized term. Students often write $- \sqrt{8} + \sqrt{7}$ instead of $- (\sqrt{8} + \sqrt{7}) = - \sqrt{8} - \sqrt{7}$. This prevents the terms from canceling out properly and leads to a chaotic, incorrect algebraic mess. Always use parentheses when substituting multi-term expressions back into an equation with subtraction. ### Final Answer **Therefore, the correct answer is 5.**