r = rac{(5 + \sqrt{5})(1 - \sqrt{5})}{(1 + \sqrt{5})(1 - \sqrt{5})} = rac{5 - 5\sqrt{5} + \sqrt{5} - 5}{1 - 5} = rac{-4\sqrt{5}}{-4} = \sqrt{5}.

["Understanding the Elegant Identity: Why r = √5 from a Simplified Radical Expression", "In mathematics, especially algebra, certain expressions may appear complex at first glance, but often simplify to beautiful, elegant results. One such identity is derived from the fraction:", "[\nr = \frac{(5 + \sqrt{5})(1 - \sqrt{5})}{(1 + \sqrt{5})(1 - \sqrt{5})}\n]", "At first glance, expanding this expression seems daunting due to nested radicals and binomials. However, with careful algebraic manipulation, we uncover a remarkably clean result:\n[\nr = \sqrt{5}\n]", "Let’s walk through the derivation step-by-step to understand how this identity emerges, revealing both computational clarity and mathematical harmony.", "---", "### Step 1: Expand the Numerator", "The numerator is ((5 + \sqrt{5})(1 - \sqrt{5})). Apply the distributive property (FOIL method):", "[\n(5 + \sqrt{5})(1 - \sqrt{5}) = 5 \cdot 1 + 5 \cdot (-\sqrt{5}) + \sqrt{5} \cdot 1 + \sqrt{5} \cdot (-\sqrt{5})\n]", "[\n= 5 - 5\sqrt{5} + \sqrt{5} - (\sqrt{5})^2\n]", "Since ((\sqrt{5})^2 = 5), this simplifies to:", "[\n5 - 5\sqrt{5} + \sqrt{5} - 5 = (-5\sqrt{5} + \sqrt{5}) + (5 - 5) = -4\sqrt{5}\n]", "---", "### Step 2: Simplify the Denominator", "The denominator is ((1 + \sqrt{5})(1 - \sqrt{5})), a classic difference of squares:", "[\n(1 + \sqrt{5})(1 - \sqrt{5}) = 1^2 - (\sqrt{5})^2 = 1 - 5 = -4\n]", "---", "### Step 3: Combine Numerator and Denominator", "Now substitute both simplified parts back:", "[\nr = \frac{-4\sqrt{5}}{-4}\n]", "Cancel the common factor of (-4) in numerator and denominator:", "[\nr = \sqrt{5}\n]", "---", "### Why This Identity Is Significant", "The result (\boxed{r = \sqrt{5}}) is more than just a numerical identity—it highlights the power of algebraic simplification. Nested irrational expressions often obscure underlying simplicity, but through systematic expansion and cancellation, we reveal clarity and elegance.", "This identity can serve educational purposes, demonstrating:", "- Mastery of binomial multiplication\n- Understanding difference of squares\n- Efficient simplification techniques\n- Confidence in manipulating radicals", "Moreover, (\sqrt{5}) is an important irrational number appearing in classical geometry (e.g., the diagonal of a unit square), number theory, and continued fractions—making this identity practically meaningful beyond mere algebra.", "---", "### Final Thoughts", "Mathematics often rewards patience and precision. The transformation from a complex-looking fraction to the simple yet profound result (\sqrt{5}) illustrates how mathematics uncovers order hidden within complexity. Whether teaching algebraic skills or deepening appreciation for mathematical beauty, expressions like this remind us that even the most intricate formulas can lead to clear, elegant truths.", "---", "Keywords: r = (5 + √5)(1 − √5) / (1 + √5)(1 − √5), simplified radical identity, √5 derivation, algebraic simplification, difference of squares, mathematics elegance, symbolic math, radical simplification."]