Question: Rationalize the denominator of $ \frac{\sqrt{5}}{\sqrt{3} - \sqrt{2}} $.
["How to Rationalize the Denominator of $ \dfrac{\sqrt{5}}{\sqrt{3} - \sqrt{2}} $: A Step-by-Step Guide", "Rationalizing the denominator is a fundamental technique in algebra, especially when working with expressions involving square roots in the denominator. One common example is simplifying fractions with radicals like $ \dfrac{\sqrt{5}}{\sqrt{3} - \sqrt{2}} $. In this article, we’ll explore how to rationalize this denominator step by step and understand why it matters.", "---", "### Why Rationalize the Denominator?", "Rationalizing the denominator means eliminating radicals from the bottom (denominator) of a fraction. This simplifies expressions, makes them easier to work with in further calculations, and aligns with standard algebraic form for many expressions.", "---", "### Step-by-Step: Rationalizing $ \dfrac{\sqrt{5}}{\sqrt{3} - \sqrt{2}} $", "The key idea is to multiply both numerator and denominator by the conjugate of the denominator. The conjugate of $ \sqrt{3} - \sqrt{2} $ is $ \sqrt{3} + \sqrt{2} $, because multiplying conjugates removes the square roots via the difference of squares formula:", "$$\n(a - b)(a + b) = a^2 - b^2\n$$", "Let’s apply this method:", "#### Step 1: Identify the conjugate\nThe denominator $ \sqrt{3} - \sqrt{2} $ has conjugate $ \sqrt{3} + \sqrt{2} $.", "#### Step 2: Multiply numerator and denominator by the conjugate\n$$\n\dfrac{\sqrt{5}}{\sqrt{3} - \sqrt{2}} \cdot \dfrac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}} = \dfrac{\sqrt{5}(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})}\n$$", "#### Step 3: Simplify the denominator\nUse the difference of squares:\n$$\n(\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1\n$$", "So the denominator simplifies to 1.", "#### Step 4: Simplify the expression\n$$\n\dfrac{\sqrt{5}(\sqrt{3} + \sqrt{2})}{1} = \sqrt{5}(\sqrt{3} + \sqrt{2})\n$$", "---", "### Final Result", "$$\n\dfrac{\sqrt{5}}{\sqrt{3} - \sqrt{2}} = \sqrt{5}(\sqrt{3} + \sqrt{2})\n$$", "Which can also be written as:", "$$\n\sqrt{15} + \sqrt{10}\n$$", "---", "### Summary", "Rationalizing the denominator of $ \dfrac{\sqrt{5}}{\sqrt{3} - \sqrt{2}} $ involves multiplying numerator and denominator by the conjugate $ \sqrt{3} + \sqrt{2} $, simplifying the denominator from $ \sqrt{3} - \sqrt{2} $ to 1, and combining the radicals in the numerator. This technique transforms an otherwise complex expression into a clean, simplified form.", "Understanding this process is essential for mastering radical expressions and algebraic manipulation, helping students and learners build a strong foundation for advanced mathematics.", "---", "Keywords: Rationalize denominator, $ \dfrac{\sqrt{5}}{\sqrt{3} - \sqrt{2}} $, simplify radicals, conjugate methods, algebra tips, square root expressions."]
