Given that $\sqrt{3} = 1.732$, the value of $$ \frac{3 + \sqrt{6}}{5\sqrt{3} - 2\sqrt{12} - \sqrt{32} + \sqrt{50}} $$ is

### Concept & Formula The core strategy is simplifying compound surds by extracting perfect square factors from inside the square roots. Use the property: $$ \sqrt{a^2 \times b} = a\sqrt{b} $$ After breaking down all terms into their simplest radical forms, combine like terms to simplify the fraction. ### Step-by-Step Solution Let's simplify the numerator and denominator separately. **Numerator:** $3 + \sqrt{6}$ We can extract a common factor of $\sqrt{3}$ because $3 = (\sqrt{3})^2$ and $\sqrt{6} = \sqrt{3}\sqrt{2}$: Numerator $= \sqrt{3}(\sqrt{3} + \sqrt{2})$ **Denominator:** $5\sqrt{3} - 2\sqrt{12} - \sqrt{32} + \sqrt{50}$ Simplify each term by factoring out perfect squares: $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$ $\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$ $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$ Substitute these back into the denominator: $= 5\sqrt{3} - 2(2\sqrt{3}) - 4\sqrt{2} + 5\sqrt{2}$ $= 5\sqrt{3} - 4\sqrt{3} - 4\sqrt{2} + 5\sqrt{2}$ Group the like terms (the $\sqrt{3}$ terms together and the $\sqrt{2}$ terms together): $= (5 - 4)\sqrt{3} + (5 - 4)\sqrt{2}$ $= \sqrt{3} + \sqrt{2}$ **Recombine Fraction:** Expression $= \frac{\sqrt{3}(\sqrt{3} + \sqrt{2})}{\sqrt{3} + \sqrt{2}}$ The term $(\sqrt{3} + \sqrt{2})$ cancels out from the top and bottom: Expression $= \sqrt{3}$ Substitute the given value $\sqrt{3} = 1.732$. ### Exam Strategy & Shortcut In competitive exams, complicated-looking surd fractions almost always simplify down to a single radical term. Once you spot that the denominator simplifies to $\sqrt{3} + \sqrt{2}$, look at the numerator $3 + \sqrt{6}$. Recognize instantly that $\sqrt{3}$ is a multiplier that maps the denominator to the numerator. Cancel them mentally to arrive at $\sqrt{3}$ directly. ### Common Pitfall A major trap is attempting to plug in $1.732$ for $\sqrt{3}$ and approximating other roots like $\sqrt{2}$ at the very beginning. This creates a nightmare of decimal arithmetic that wastes time and guarantees calculation errors. Always simplify variables and radicals fully before substituting numeric values. ### Final Answer **Therefore, the correct answer is 1.732.**