$\frac{12}{3 + \sqrt{5} + 2\sqrt{2}}$ is equal to

### Concept & Formula When a denominator contains three surd terms, you must rationalize it by strategically grouping the terms into two parts to create a binomial structure, and then multiplying by its conjugate. $$ (a+b)(a-b) = a^2 - b^2 $$ ### Step-by-Step Solution **Given:** The algebraic fraction $\frac{12}{3 + \sqrt{5} + 2\sqrt{2}}$ **Calculation / Deduction:** * Group the denominator into a single binomial structure to prepare for rationalization: $((3 + \sqrt{5}) + 2\sqrt{2})$ * Rationalize by multiplying the numerator and the denominator by the conjugate $((3 + \sqrt{5}) - 2\sqrt{2})$: Denominator $= (3 + \sqrt{5})^2 - (2\sqrt{2})^2$ * Expand the first term of the denominator using $(a+b)^2 = a^2 + 2ab + b^2$: Denominator $= (9 + 5 + 6\sqrt{5}) - 8 = 14 + 6\sqrt{5} - 8 = 6 + 6\sqrt{5} = 6(1 + \sqrt{5})$ * Update the entire fraction with the simplified denominator: Fraction $= \frac{12(3 + \sqrt{5} - 2\sqrt{2})}{6(1 + \sqrt{5})} = \frac{2(3 + \sqrt{5} - 2\sqrt{2})}{1 + \sqrt{5}}$ * The denominator still contains a surd, so rationalize again by multiplying the top and bottom by $(\sqrt{5} - 1)$: New Denominator $= (\sqrt{5})^2 - (1)^2 = 5 - 1 = 4$ * Multiply the numerator expressions out carefully: Numerator $= 2(3 + \sqrt{5} - 2\sqrt{2})(\sqrt{5} - 1)$ $= 2(3\sqrt{5} - 3 + 5 - \sqrt{5} - 2\sqrt{10} + 2\sqrt{2})$ $= 2(2 + 2\sqrt{5} + 2\sqrt{2} - 2\sqrt{10})$ Factor out the common $2$: Numerator $= 4(1 + \sqrt{5} + \sqrt{2} - \sqrt{10})$ * Divide the final numerator by the new denominator ($4$): Result $= \frac{4(1 + \sqrt{5} + \sqrt{2} - \sqrt{10})}{4} = 1 + \sqrt{5} + \sqrt{2} - \sqrt{10}$ ### Exam Strategy & Shortcut For three-term rationalization, always group the two terms whose squares sum closest to the square of the third term. Here, $3^2 + (\sqrt{5})^2 = 9 + 5 = 14$, and $(2\sqrt{2})^2 = 8$. This keeps the resulting coefficients manageable. Keep constants factored out (like keeping the 12 outside the brackets) until the very end to maximize cancellation opportunities. ### Common Pitfall Students often distribute the numerator's constant immediately instead of looking for structural cancellations. Distributing the $12$ into the numerator right away makes the subsequent multiplication by $(\sqrt{5} - 1)$ much harder and severely increases the likelihood of simple arithmetic errors. ### Final Answer **Therefore, the correct answer is $1 + \sqrt{5} + \sqrt{2} - \sqrt{10}$.**