5Question: Rationalize the denominator of $\displaystyle rac{3}{\sqrt{7} - \sqrt{2}}$ and simplify.

["Title: Rationalizing the Denominator: A Step-by-Step Guide to Simplifying $\displaystyle \frac{3}{\sqrt{7} - \sqrt{2}}$", "Introduction\nWhen working with radicals in algebra, one essential skill is rationalizing the denominator. Whether solving equations, simplifying expressions, or preparing for integration or limits, knowing how to eliminate radicals from the denominator ensures clarity and mathematical precision. In this article, we’ll walk through the process using a classic example: rationalizing $\displaystyle \frac{3}{\sqrt{7} - \sqrt{2}}$, and explain how to simplify it thoroughly.", "---", "Why Rationalize the Denominator?\nRationalizing the denominator removes irrational numbers from the bottom of a fraction. This practice standardizes expressions, improves readability, and is often required in advanced math, engineering, and scientific calculations. For the expression $\displaystyle \frac{3}{\sqrt{7} - \sqrt{2}}$, the denominator $\sqrt{7} - \sqrt{2}$ contains two square roots, making it less clean and more difficult to work with directly.", "---", "The Rationalization Method\nTo rationalize a denominator involving a binomial with square roots, we multiply both the numerator and the denominator by the conjugate of the denominator.", "The conjugate of $\sqrt{7} - \sqrt{2}$ is $\sqrt{7} + \sqrt{2}$.\nMultiplying by this conjugate eliminates the square roots via the difference of squares formula:\n$$\n(a - b)(a + b) = a^2 - b^2\n$$", "Apply this to the denominator:", "$$\n(\sqrt{7} - \sqrt{2})(\sqrt{7} + \sqrt{2}) = (\sqrt{7})^2 - (\sqrt{2})^2 = 7 - 2 = 5\n$$", "---", "Step-by-Step Simplification", "Start with:\n$$\n\frac{3}{\sqrt{7} - \sqrt{2}}\n$$", "Multiply numerator and denominator by the conjugate $\sqrt{7} + \sqrt{2}$:", "$$\n\frac{3}{\sqrt{7} - \sqrt{2}} \cdot \frac{\sqrt{7} + \sqrt{2}}{\sqrt{7} + \sqrt{2}} = \frac{3(\sqrt{7} + \sqrt{2})}{(\sqrt{7} - \sqrt{2})(\sqrt{7} + \sqrt{2})}\n$$", "Simplify the denominator using the difference of squares:", "$$\n= \frac{3(\sqrt{7} + \sqrt{2})}{7 - 2} = \frac{3(\sqrt{7} + \sqrt{2})}{5}\n$$", "---", "Final Simplified Form", "$$\n\frac{3(\sqrt{7} + \sqrt{2})}{5} = \frac{3\sqrt{7} + 3\sqrt{2}}{5}\n$$", "---", "Conclusion\nRationalizing the denominator of $\displaystyle \frac{3}{\sqrt{7} - \sqrt{2}}$ transforms it into a simplified, clean expression:\n$$\n\boxed{\frac{3\sqrt{7} + 3\sqrt{2}}{5}}\n$$", "This form is not only mathematically precise but also easier to evaluate, differentiate, or integrate. Mastering rationalization steps like identifying conjugates, applying algebraic identities, and simplifying fully is key to working confidently with radicals in algebra and beyond.", "---", "Keywords: rationalize denominator, simplify radicals, rationalization of European form, rationalize $\displaystyle \frac{3}{\sqrt{7} - \sqrt{2}}$, algebra techniques, difference of squares, simplify $\displaystyle \frac{3}{\sqrt{7} - \sqrt{2}}$, step-by-step rationalization."]