\frac{1 + \frac{\sqrt{7}}{3}}{1 - \sqrt{7}} = \frac{\frac{3 + \sqrt{7}}{3}}{1 - \sqrt{7}} = \frac{3 + \sqrt{7}}{3(1 - \sqrt{7})}.
["Simplifying the Expression: How to Properly Rationalize and Evaluate (\frac{1 + \frac{\sqrt{7}}{3}}{1 - \sqrt{7}})", "When dealing with algebraic expressions involving square roots, one common yet essential operation is rationalization — particularly when denominators contain irrational numbers. In this article, we explore the step-by-step simplification of the equation:", "[\n\frac{1 + \frac{\sqrt{7}}{3}}{1 - \sqrt{7}} = \frac{\frac{3 + \sqrt{7}}{3}}{1 - \sqrt{7}} = \frac{3 + \sqrt{7}}{3(1 - \sqrt{7})}\n]", "Understanding how to manipulate such expressions is crucial for solving equations, simplifying fractions with radicals, and preparing for advanced algebra or calculus problems.", "---", "### Step 1: Rewriting the Original Expression", "We begin with:", "[\n\frac{1 + \frac{\sqrt{7}}{3}}{1 - \sqrt{7}}\n]", "To simplify, combine the terms in the numerator:", "[\n1 + \frac{\sqrt{7}}{3} = \frac{3}{3} + \frac{\sqrt{7}}{3} = \frac{3 + \sqrt{7}}{3}\n]", "Substituting back gives:", "[\n\frac{\frac{3 + \sqrt{7}}{3}}{1 - \sqrt{7}} = \frac{3 + \sqrt{7}}{3(1 - \sqrt{7})}\n]", "This expression is simpler and standardizes the form for further rationalization.", "---", "### Step 2: Rationalizing the Denominator", "The denominator (1 - \sqrt{7}) contains an irrational number. To rationalize, multiply numerator and denominator by the conjugate of the denominator — in this case, (1 + \sqrt{7}):", "[\n\frac{3 + \sqrt{7}}{3(1 - \sqrt{7})} \cdot \frac{1 + \sqrt{7}}{1 + \sqrt{7}} = \frac{(3 + \sqrt{7})(1 + \sqrt{7})}{3(1 - \sqrt{7})(1 + \sqrt{7})}\n]", "---", "### Step 3: Expand Numerator and Denominator", "Denominator:\nUse the difference of squares:\n[\n(1 - \sqrt{7})(1 + \sqrt{7}) = 1^2 - (\sqrt{7})^2 = 1 - 7 = -6\n]\nThus, denominator becomes:\n[\n3(-6) = -18\n]", "Numerator:\nExpand ((3 + \sqrt{7})(1 + \sqrt{7})):", "[\n3 \cdot 1 + 3 \cdot \sqrt{7} + \sqrt{7} \cdot 1 + \sqrt{7} \cdot \sqrt{7} = 3 + 3\sqrt{7} + \sqrt{7} + 7 = 10 + 4\sqrt{7}\n]", "So numerator is:\n[\n10 + 4\sqrt{7}\n]", "---", "### Step 4: Combine and Simplify", "Putting it all together:", "[\n\frac{10 + 4\sqrt{7}}{-18} = -\frac{10 + 4\sqrt{7}}{18}\n]", "We can factor numerator and simplify:", "[\n-\frac{2(5 + 2\sqrt{7})}{18} = -\frac{5 + 2\sqrt{7}}{9}\n]", "---", "### Final Simplified Form", "Thus, the fully simplified and rationalized form is:", "[\n\frac{1 + \frac{\sqrt{7}}{3}}{1 - \sqrt{7}} = \frac{3 + \sqrt{7}}{3(1 - \sqrt{7})} = \boxed{-\frac{5 + 2\sqrt{7}}{9}}\n]", "---", "### Why Rationalization Matters", "Rationalizing denominators eliminates irrational numbers in denominators, which is important not only for aesthetic clarity but also for numerical approximation, integration, and ensuring expressions are in the preferred form for further computation. Mastering this technique supports better mastery of algebraic expressions and problem-solving in advanced math.", "---", "Keywords for SEO Optimization:\n- Simplify (\frac{1 + \frac{\sqrt{7}}{3}}{1 - \sqrt{7}})\n- Rationalizing denominator with square roots\n- Simplify radical expressions\n- Solve (\frac{3 + \sqrt{7}}{3(1 - \sqrt{7})})\n- Algebraic simplification with square roots\n- Step-by-step rationalization process", "---", "By following these steps, you can confidently transform complex fractional expressions into their simplest and most usable forms — a cornerstone skill for algebra success."]
