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

["Simplifying and Rationalizing a Complex Expression: A Step-by-Step Breakdown of $ r = \frac{5 + \sqrt{5}}{1 + \sqrt{5}} \cdot \frac{1 - \sqrt{5}}{1 - \sqrt{5}} $", "In algebra, simplifying radical expressions and rationalizing denominators are essential techniques that enhance clarity and pave the way for further manipulation. One such fascinating expression is:", "$$\nr = \frac{5 + \sqrt{5}}{1 + \sqrt{5}} \cdot \frac{1 - \sqrt{5}}{1 - \sqrt{5}}\n$$", "At first glance, this looks complex, but with careful step-by-step simplification and rationalization, we can transform it into a fully simplified real expression.", "---", "### Step 1: Multiply the Fractions", "We begin by combining the two fractions under a single fraction:", "$$\nr = \frac{(5 + \sqrt{5})(1 - \sqrt{5})}{(1 + \sqrt{5})(1 - \sqrt{5})}\n$$", "This leverages the identity $ (a + b)(a - b) = a^2 - b^2 $, provided the denominator is a difference of squares.", "---", "### Step 2: Simplify the Denominator", "The denominator is:", "$$\n(1 + \sqrt{5})(1 - \sqrt{5}) = 1^2 - (\sqrt{5})^2 = 1 - 5 = -4\n$$", "This gives us:", "$$\nr = \frac{(5 + \sqrt{5})(1 - \sqrt{5})}{-4}\n$$", "---", "### Step 3: Expand the Numerator", "Now expand $ (5 + \sqrt{5})(1 - \sqrt{5}) $ using the distributive property:", "$$\n(5 + \sqrt{5})(1 - \sqrt{5}) = 5 \cdot 1 - 5 \cdot \sqrt{5} + \sqrt{5} \cdot 1 - \sqrt{5} \cdot \sqrt{5}\n$$\n$$\n= 5 - 5\sqrt{5} + \sqrt{5} - 5\n$$", "Simplify:", "$$\n(5 - 5) + (-5\sqrt{5} + \sqrt{5}) = 0 - 4\sqrt{5} = -4\sqrt{5}\n$$", "---", "### Step 4: Final Simplification", "Substitute the simplified numerator back:", "$$\nr = \frac{-4\sqrt{5}}{-4} = \sqrt{5}\n$$", "---", "### Conclusion", "The original expression simplifies elegantly to:", "$$\n\boxed{r = \sqrt{5}}\n$$", "This demonstrates how rationalizing denominators and carefully expanding products with radicals leads to a clean, simplified radical expression. Though the initial form appeared algebraically dense, step-by-step simplification reveals a concise result. Whether preparing for advanced math problems or mastering algebraic techniques, mastering such rationalization and simplification is invaluable.", "---", "Key Takeaways:", "- Recognize the difference of squares to simplify denominators.\n- Carefully expand binomial products involving radicals.\n- Combine numerator and denominator cleanly after manipulation.\n- Rationalization and simplification enhance understanding and usability in further analysis.", "Keywords:\n$ \frac{5 + \sqrt{5}}{1 + \sqrt{5}} \cdot \frac{1 - \sqrt{5}}{1 - \sqrt{5}} $, rationalizing denominator, simplifying radicals, algebraic expression, simplification steps, difference of squares, algebraic identity, nested radicals, mathematical simplification."]