To rationalize the denominator, we multiply both numerator and denominator by the conjugate of the denominator, which is $\sqrt{7} + \sqrt{2}$:

["Rationalizing Denominators Using Conjugates: A Step-by-Step Guide", "In algebra, simplifying expressions is essential for clarity and further manipulation. One common task is rationalizing the denominator of a fraction, especially when the denominator is an expression involving square roots, such as $\sqrt{7} + \sqrt{2}$. A powerful and widely used method to eliminate radicals from the denominator is multiplying both the numerator and denominator by the conjugate of the denominator. In this article, we’ll explore how to rationalize the denominator by multiplying by $\sqrt{7} + \sqrt{2}$, why this technique works, and a clear example to illustrate the process.", "---", "### What Is a Conjugate?", "For any binomial involving square roots, the conjugate is formed by changing the sign between the two radicals. For the expression $\sqrt{7} + \sqrt{2}$, its conjugate is $\sqrt{7} - \sqrt{2}$. The purpose of multiplying by the conjugate is to eliminate the square roots in the denominator through a clever algebraic identity.", "---", "### Why Multiply by the Conjugate?", "When a binomial with square roots is multiplied by its conjugate, the result simplifies significantly thanks to the difference of squares formula:", "$$\n(a + b)(a - b) = a^2 - b^2\n$$", "Apply this to the denominator:", "$$\n(\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2}) = (\sqrt{7})^2 - (\sqrt{2})^2 = 7 - 2 = 5\n$$", "So the denominator becomes a rational number, simplifying the original fraction.", "---", "### Step-by-Step: Rationalizing $\frac{1}{\sqrt{7} + \sqrt{2}}$", "Let’s apply the method to a concrete example. Rationalize the denominator of:", "$$\n\frac{1}{\sqrt{7} + \sqrt{2}}\n$$", "#### Step 1: Identify the conjugate\nThe conjugate of $\sqrt{7} + \sqrt{2}$ is $\sqrt{7} - \sqrt{2}$.", "#### Step 2: Multiply numerator and denominator by the conjugate\n$$\n\frac{1}{\sqrt{7} + \sqrt{2}} \cdot \frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} - \sqrt{2}} = \frac{\sqrt{7} - \sqrt{2}}{(\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2})}\n$$", "#### Step 3: Simplify the denominator\n$$\n(\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2}) = 7 - 2 = 5\n$$", "#### Step 4: Write the final expression\n$$\n\frac{\sqrt{7} - \sqrt{2}}{5}\n$$", "The denominator is now rational, and the expression is simplified.", "---", "### Benefits of Rationalizing the Denominator", "- Simplifies further calculations: Rational denominators are easier to work with in complex equations or integrals.\n- Standardizes mathematical presentation: Many textbooks and practical apps prefer rationalized forms.\n- Avoids irrational numbers in denominators: Industrial calculations and engineering equations often avoid irrational denominators for clarity and precision.", "---", "### Summary", "To rationalize a denominator with two square roots, multiply both numerator and denominator by their conjugate. For $\frac{1}{\sqrt{7} + \sqrt{2}}$, the conjugate is $\sqrt{7} - \sqrt{2}$. Multiplying results in a denominator of $5$ and a simplified form—crucial for clean, usable algebraic expressions.", "Whether in homework, exam problems, or real-world calculations, mastering this technique ensures smoother, more professional mathematical expressions.", "---", "Keywords for SEO:\nrationalize denominator, conjugate method, simplify radicals, rationalize $\sqrt{7} + \sqrt{2}$, algebra technique, conjugate multiplication, simplify fractions with radicals, difference of squares in algebra.", "---", "Rationalizing denominators using conjugates is not just a mechanical rule—it’s a foundational skill that builds confidence and clarity in algebra. Start practicing today to master this essential technique!"]