5Question: Rationalize the denominator of $\displaystyle rac{3}{\sqrt{7} - \sqrt{2}}$.
["Rationalizing the Denominator: Simplifying $\displaystyle \dfrac{3}{\sqrt{7} - \sqrt{2}}$", "When working with expressions that contain square roots in the denominator, it’s important to rationalize the denominator for clearer analysis and easier computation. In this article, we’ll explore step-by-step how to rationalize\n$$\n\displaystyle \dfrac{3}{\sqrt{7} - \sqrt{2}}\n$$\nand simplify the expression.", "---", "### What Does Rationalizing the Denominator Mean?", "Rationalizing the denominator means eliminating any irrational numbers (such as square roots) from the denominator of a fraction. Rational denominators improve readability and facilitate further algebraic manipulation or integration in more advanced applications.", "---", "### Step 1: Identify the Conjugate", "The denominator is $\sqrt{7} - \sqrt{2}$. To eliminate the subtraction, we multiply numerator and denominator by its conjugate, which replaces the sign between the two radicals.", "The conjugate of $\sqrt{7} - \sqrt{2}$ is $\sqrt{7} + \sqrt{2}$.", "---", "### Step 2: Multiply Top and Bottom by the Conjugate", "We multiply both numerator and denominator by $\sqrt{7} + \sqrt{2}$:", "$$\n\dfrac{3}{\sqrt{7} - \sqrt{2}} \cdot \dfrac{\sqrt{7} + \sqrt{2}}{\sqrt{7} + \sqrt{2}} = \dfrac{3(\sqrt{7} + \sqrt{2})}{(\sqrt{7} - \sqrt{2})(\sqrt{7} + \sqrt{2})}\n$$", "---", "### Step 3: Apply the Difference of Squares Formula", "Using the identity $(a - b)(a + b) = a^2 - b^2$, the denominator becomes:", "$$\n(\sqrt{7})^2 - (\sqrt{2})^2 = 7 - 2 = 5\n$$", "So the expression simplifies to:", "$$\n\dfrac{3(\sqrt{7} + \sqrt{2})}{5}\n$$", "---", "### Final Simplified Expression", "$$\n\dfrac{3(\sqrt{7} + \sqrt{2})}{5}\n$$", "This is the fully rationalized form of the original fraction.", "---", "### Why Rationalize?", "- Improves clarity and usability in further computations.\n- Removes irrational numbers from the denominator, which is often a requirement in algebra and calculus.\n- Helps prevent errors in numerical approximation and symbolic analysis.", "---", "### Summary", "To rationalize $\displaystyle \dfrac{3}{\sqrt{7} - \sqrt{2}}$, we multiplied numerator and denominator by the conjugate $\sqrt{7} + \sqrt{2}$, applied the difference of squares identity, and simplified the resulting expression:", "$$\n\dfrac{3(\sqrt{7} + \sqrt{2})}{5}\n$$", "This technique is essential for clean, professional, and accurate mathematical handling of irrational denominators.", "---", "Keywords: rationalize denominator, simplify expression, rationalize $\dfrac{3}{\sqrt{7} - \sqrt{2}}$, algebraic manipulation, difference of squares, math tutorial, rationalize radicals, algebra practice."]
