\[ \frac{3 + \sqrt{5}}{2 - \sqrt{5}} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}} = \frac{(3 + \sqrt{5})(2 + \sqrt{5})}{(2 - \sqrt{5})(2 + \sqrt{5})} \]

["# Simplifying Radical Expressions: A Step-by-Step Guide to\n[ \frac{3 + \sqrt{5}}{2 - \sqrt{5}} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}} = \frac{(3 + \sqrt{5})(2 + \sqrt{5})}{(2 - \sqrt{5})(2 + \sqrt{5})} ]", "When working with expressions that include irrational numbers like (\sqrt{5}), rationalization plays a critical role in simplifying and evaluating complex fractions. In this article, we explore the simplification of the expression:\n[\n\frac{3 + \sqrt{5}}{2 - \sqrt{5}} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}} = \frac{(3 + \sqrt{5})(2 + \sqrt{5})}{(2 - \sqrt{5})(2 + \sqrt{5})}\n]\nand demonstrate why rationalizing both numerator and denominator is a powerful technique—especially when denominators contain square roots.", "---", "## Understanding the Structure", "The given expression is a product of two fractions. Multiplying fractions allows us to combine numerators and denominators:", "[\n\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}\n]", "Applying this, we define:\n- ( a = 3 + \sqrt{5} )\n- ( b = 2 - \sqrt{5} )\n- ( c = 2 + \sqrt{5} )\n- ( d = 2 + \sqrt{5} )", "Thus,\n[\n\frac{3 + \sqrt{5}}{2 - \sqrt{5}} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}} = \frac{(3 + \sqrt{5})(2 + \sqrt{5})}{(2 - \sqrt{5})(2 + \sqrt{5})}\n]", "---", "## Rationalizing the Denominator", "Notice that the denominator contains a difference involving a square root: (2 - \sqrt{5}). To simplify, we rationalize this term by multiplying numerator and denominator by the conjugate of the denominator:\n[\n\ ext{Conjugate of } (2 - \sqrt{5}) \ ext{ is } (2 + \sqrt{5})\n]", "So we multiply both the numerator and denominator by (2 + \sqrt{5}):", "[\n\frac{(3 + \sqrt{5})(2 + \sqrt{5})}{(2 - \sqrt{5})(2 + \sqrt{5})} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}} = \frac{(3 + \sqrt{5})(2 + \sqrt{5}) \cdot (2 + \sqrt{5})}{(2 - \sqrt{5})(2 + \sqrt{5})(2 + \sqrt{5})}\n]", "However, if instead of re-rationalizing fully shown, we recognize a key simplification, we can proceed strategically:", "---", "## Observing Perfect Square Patterns", "Let’s simplify directly using conjugates:", "First, examine the denominator:\n[\n(2 - \sqrt{5})(2 + \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1\n]", "So the denominator simplifies nicely to (-1):", "[\n(2 - \sqrt{5})(2 + \sqrt{5}) = -1\n]", "Now simplify the numerator:\n[\n(3 + \sqrt{5})(2 + \sqrt{5})\n]", "Use distributive property (FOIL):\n[\n= 3 \cdot 2 + 3 \cdot \sqrt{5} + \sqrt{5} \cdot 2 + \sqrt{5} \cdot \sqrt{5}\n= 6 + 3\sqrt{5} + 2\sqrt{5} + 5\n= 11 + 5\sqrt{5}\n]", "So the entire expression becomes:", "[\n\frac{11 + 5\sqrt{5}}{2 - \sqrt{5}} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}} = \frac{11 + 5\sqrt{5}}{(2 - \sqrt{5})(2 + \sqrt{5})} = \frac{11 + 5\sqrt{5}}{-1} = - (11 + 5\sqrt{5})\n]", "But wait — the original expression was multiplied by (\frac{2 + \sqrt{5}}{2 + \sqrt{5}}), which is equal to 1 only when defined. But in fact, multiplying by 1 does nothing:", "[\n\frac{3 + \sqrt{5}}{2 - \sqrt{5}} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}} = \frac{3 + \sqrt{5}}{2 - \sqrt{5}}\n]\nunless the second fraction is meant to cancel—however, note that:", "[\n(2 + \sqrt{5}) \ ext{ appears in both numerator and denominator, but not in a direct cancellation unless properly grouped.", "Let’s revise the interpretation:\nThe expression is:", "[\n\frac{3 + \sqrt{5}}{2 - \sqrt{5}} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}}\n]", "Since (\frac{2 + \sqrt{5}}{2 + \sqrt{5}} = 1), this simplifies to:", "[\n\frac{3 + \sqrt{5}}{2 - \sqrt{5}}\n]", "But the problem presents this full as a rational expression over rationalized form. So more accurately, the expression:", "[\n\frac{(3 + \sqrt{5})(2 + \sqrt{5})}{(2 - \sqrt{5})(2 + \sqrt{5})}\n]", "Now substitute denominator:", "[\n(2 - \sqrt{5})(2 + \sqrt{5}) = -1\n]", "So:", "[\n\frac{(3 + \sqrt{5})(2 + \sqrt{5})}{-1} = - (3 + \sqrt{5})(2 + \sqrt{5})\n]", "Now expand numerator:", "[\n(3 + \sqrt{5})(2 + \sqrt{5}) = 6 + 3\sqrt{5} + 2\sqrt{5} + 5 = 11 + 5\sqrt{5}\n]", "Therefore, the full expression simplifies to:", "[\n- (11 + 5\sqrt{5}) = -11 - 5\sqrt{5}\n]", "---", "## Why Rationalizing Matters", "This problem demonstrates a fundamental principle:\n- Multiplying numerator and denominator by the conjugate eliminates radical expressions in the denominator, turning them into rational numbers.\n- Even when only one square root appears in numerator and denominator (like (2 + \sqrt{5})), rationalizing each conjugate pair (when paired) simplifies the expression completely.", "Though (\frac{2 + \sqrt{5}}{2 + \sqrt{5}}) is 1 in value, rationalizing in product form ensures clarity and correctness, especially in algebraic computation and equation solving.", "---", "## Final Simplified Value", "Thus,\n[\n\frac{3 + \sqrt{5}}{2 - \sqrt{5}} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}} = - (11 + 5\sqrt{5})\n]", "This form is simplified, rational, and free of radical denominators—ideal for further algebraic manipulation or numerical approximation.", "---", "## Conclusion", "Working with radical expressions can be simplified elegantly using conjugates and rationalization. This example illustrates how transforming complex fractions using strategic multiplication leads to clean, rational results. Whether solving equations, computing limits, or proving identities, mastering rationalization techniques empowers deeper comprehension and error-free computation.", "Key Takeaways:\n- Always identify conjugates to simplify irrational denominators.\n- Rationalization isn’t just cosmetic—it simplifies future operations.\n- Even repeated factors cancel, but rational form ensures correctness in mathematical rigor.", "---", "Keywords:\n[ \frac{3 + \sqrt{5}}{2 - \sqrt{5}} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}} = \frac{(3 + \sqrt{5})(2 + \sqrt{5})}{(2 - \sqrt{5})(2 + \sqrt{5})} ], simplify radical expression, rationalizing denominator, algebra simplification, conjugate multiplication, rational number evaluation, irrational algebra.", "---", "Meta Title:\nSimplifying (\frac{3 + \sqrt{5}}{2 - \sqrt{5}} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}}) — Step-by-Step Rationalization Guide", "Meta Description:\nLearn how to rationalize and simplify expressions involving radicals. This article breaks down the step-by-step simplification of (\frac{3 + \sqrt{5}}{2 - \sqrt{5}} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}}), showing why rationalizing eliminates irrational denominators and yields (- (11 + 5\sqrt{5})).", "---", "Read more about algebraic simplification techniques on our math learning hub.**"]