Question: Rationalize the denominator of $ rac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} $.
["Title: How to Rationalize the Denominator: A Step-by-Step Guide with $ \dfrac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} $", "---", "### Understanding Rationalizing the Denominator", "When solving mathematical expressions involving radicals, one common task is rationalizing the denominator. This means eliminating any irrational numbers—especially square roots—from the denominator of a fraction. Rationalizing makes expressions cleaner, simplifies calculations, and is often required in algebra and advanced math.", "In this article, we’ll learn how to rationalize the denominator of the expression:", "$$\n\dfrac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\n$$", "We’ll break down the entire process, apply the technique rigorously, and explore its significance.", "---", "### Step 1: Identify the Irrational Denominator", "The expression has the form:", "$$\n\frac{a + b}{a - b}\n\quad \ ext{where} \quad a = \sqrt{7},\quad b = \sqrt{3}\n$$", "The denominator is $ \sqrt{7} - \sqrt{3} $, which contains two square roots. Multiplying numerator and denominator by the conjugate of the denominator removes the square roots from the denominator.", "---", "### Step 2: Multiply by the Conjugate", "The conjugate of $ \sqrt{7} - \sqrt{3} $ is $ \sqrt{7} + \sqrt{3} $. To rationalize, multiply both the numerator and denominator by this conjugate:", "$$\n\dfrac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} \ imes \dfrac{\sqrt{7} + \sqrt{3}}{\sqrt{7} + \sqrt{3}} = \frac{(\sqrt{7} + \sqrt{3})^2}{(\sqrt{7} - \sqrt{3})(\sqrt{7} + \sqrt{3})}\n$$", "---", "### Step 3: Simplify the Denominator", "Use the difference of squares formula:\n$$\n(a - b)(a + b) = a^2 - b^2\n$$", "Apply to the denominator:", "$$\n(\sqrt{7})^2 - (\sqrt{3})^2 = 7 - 3 = 4\n$$", "So the denominator simplifies nicely to 4.", "---", "### Step 4: Expand the Numerator", "Now expand $ (\sqrt{7} + \sqrt{3})^2 $:", "$$\n(\sqrt{7} + \sqrt{3})^2 = (\sqrt{7})^2 + 2\sqrt{7}\sqrt{3} + (\sqrt{3})^2 = 7 + 2\sqrt{21} + 3 = 10 + 2\sqrt{21}\n$$", "---", "### Step 5: Combine Results", "Now substitute both simplified numerator and denominator:", "$$\n\frac{10 + 2\sqrt{21}}{4}\n$$", "Factor numerator for simpler expression:", "$$\n= \frac{2(5 + \sqrt{21})}{4} = \frac{5 + \sqrt{21}}{2}\n$$", "---", "### Final Answer", "$$\n\boxed{\frac{5 + \sqrt{21}}{2}}\n$$", "This is the fully simplified, rationalized form of the original expression.", "---", "### Why Rationalizing Denominators Matters", "- Simplifies expressions: Eliminates radicals from denominators, making the expression cleaner and easier to work with.\n- Standard form: Many textbooks and calculators prefer or require rationalized denominators.\n- Facilitates further operations: Useful in integration, limits, and algebraic manipulations.", "---", "### Key Takeaway", "To rationalize a denominator like $ \sqrt{a} \pm \sqrt{b} $, multiply numerator and denominator by the conjugate $ \sqrt{a} \pm \sqrt{b} $, leverage the difference of squares formula in the denominator, and simplify the resulting radical terms.", "---", "Keywords: Rationalize denominator, simplify radical expression, rationalize denominator steps, rationalizing $ \dfrac{\sqrt{7}+\sqrt{3}}{\sqrt{7}-\sqrt{3}} $, step-by-step solution, algebra tip, math tutorial.", "---", "Perfect for students mastering algebra, pre-calculus, and college-level math—this method ensures clarity and precision when working with irrational denominators."]
