$$ \frac{\sqrt{5}(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} $$
["### Simplify $$ \frac{\sqrt{5}(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} $$ – Step-by-Step Algebraic Breakdown & Simplification", "If you’ve stumbled across the expression\n$$ \frac{\sqrt{5}(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})}, $$\nyou’re not alone—this is a classic example of algebraic manipulation involving radicals and rationalization. In this article, we’ll break down the simplification step-by-step, explain key algebraic principles, and show how to simplify this expression efficiently. Whether you’re a student, educator, or math enthusiast, understanding this expression improves both algebraic fluency and confidence in working with radicals.", "---", "#### Step 1: Recognize a Key Algebraic Identity", "The denominator $$ (\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) $$ is a difference of squares, which follows the identity:\n$$\n(a - b)(a + b) = a^2 - b^2\n$$\nHere, $ a = \sqrt{3} $ and $ b = \sqrt{2} $. Applying the identity:\n$$\n(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) = (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1\n$$", "This dramatically simplifies the denominator from a binomial product to just 1.", "---", "#### Step 2: Rewrite the Full Expression Using the Simplified Denominator", "Substituting the simplified denominator back into the original expression:\n$$\n\frac{\sqrt{5}(\sqrt{3} + \sqrt{2})}{1} = \sqrt{5}(\sqrt{3} + \sqrt{2})\n$$", "---", "#### Step 3: (Optional) Distribute and Final Simplified Form", "While not necessary, distributing $ \sqrt{5} $ gives:\n$$\n\sqrt{5}\sqrt{3} + \sqrt{5}\sqrt{2} = \sqrt{15} + \sqrt{10}\n$$\nSo the fully simplified and factored form is:\n$$\n\sqrt{15} + \sqrt{10}\n$$", "---", "### Why Is This Simplification Important?", "- Rationalizing Denominators: Though this expression already has a rational denominator, recognizing when and how to rationalize is foundational in algebra and calculus.\n- Efficiency: Recognizing the difference of squares avoids unnecessary complexity and speeds up computation.\n- Pattern Recognition: Practicing such expressions builds intuition for handling nested radicals and fractional exponents.", "---", "### Key Takeaways", "- Use the identity $ (a - b)(a + b) = a^2 - b^2 $ to simplify products involving square roots.\n- Never underestimate the power of simplifying denominators—sometimes a product yields 1!\n- The simplification $ \frac{\sqrt{5}(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} = \sqrt{15} + \sqrt{10} $ demonstrates clean transformation using basic algebra and radical rules.", "---", "### Final Answer", "$$\n\boxed{\sqrt{15} + \sqrt{10}}\n$$", "---\nKeywords for SEO:\n$$ \frac{\sqrt{5}(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})}, \ ext{ simplify radical expression, rationalize denominator, algebraic simplification, step-by-step Radical simplification, delta of squares, algebraic identities, math practice, rationalizing radicals, simplify } \sqrt{5}(\sqrt{3} + \sqrt{2}) $$", "---", "Elevate your algebra skills by mastering expressions like this one—one step, one identity at a time!"]
