Solution: To rationalize the denominator, multiply numerator and denominator by the conjugate $\sqrt{7} + \sqrt{2}$:
["Solution: Rationalizing the Denominator Using the Conjugate $ \sqrt{7} + \sqrt{2} $", "When working with algebraic expressions involving square roots in the denominator, rationalizing the denominator is a crucial step to simplify expressions and make them easier to work with—whether for handwritten homework, mathematical exams, or advanced problem-solving. One of the most effective methods to rationalize a denominator containing a binomial with square roots is to multiply both the numerator and the denominator by the conjugate of the denominator.", "### Understanding the Problem", "Consider the common expression:", "[\n\frac{1}{\sqrt{7} + \sqrt{2}}\n]", "Here, the denominator is a sum of two square roots. Directly working with such a denominator can complicate further calculations. Rationalization transforms it into a simpler, rational expression by eliminating the radicals from the denominator.", "### Why Use the Conjugate?", "The conjugate of a binomial $ a + b $ is $ a - b $. When multiplied, it eliminates the radicals due to the difference of squares formula:", "[\n(a + b)(a - b) = a^2 - b^2\n]", "In this case, with $ a = \sqrt{7} $ and $ b = \sqrt{2} $, the conjugate is:", "[\n\sqrt{7} - \sqrt{2}\n]", "Multiplying numerator and denominator by $ \sqrt{7} - \sqrt{2} $ allows us to eliminate the radicals in the denominator.", "### Step-by-Step Rationalization", "Start with:", "[\n\frac{1}{\sqrt{7} + \sqrt{2}}\n]", "Multiply both numerator and denominator by the conjugate $ \sqrt{7} - \sqrt{2} $:", "[\n\frac{1}{\sqrt{7} + \sqrt{2}} \cdot \frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} - \sqrt{2}} = \frac{\sqrt{7} - \sqrt{2}}{(\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2})}\n]", "Now simplify the denominator using the difference of squares:", "[\n(\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2}) = (\sqrt{7})^2 - (\sqrt{2})^2 = 7 - 2 = 5\n]", "So the expression simplifies to:", "[\n\frac{\sqrt{7} - \sqrt{2}}{5}\n]", "### Final Rationalized Form", "[\n\boxed{\frac{\sqrt{7} - \sqrt{2}}{5}}\n]", "This final form is fully simplified and has a rational denominator, making it more usable in further mathematical operations.", "### Summary", "- Problem: Denominator contains a binomial with square roots: $ \sqrt{7} + \sqrt{2} $\n- Solution: Multiply numerator and denominator by the conjugate: $ \sqrt{7} - \sqrt{2} $\n- Why it works: The product of conjugates yields a difference of squares, eliminating the radicals\n- Result: Simplified, rationalized expression: $ \dfrac{\sqrt{7} - \sqrt{2}}{5} $", "Rationalizing denominators using conjugates is a fundamental technique that enhances clarity and precision in algebra—essential for mastering more complex mathematical concepts."]
