\frac{a + \frac{1}{2} \cdot \frac{2a\sqrt{7}}{3}}{a - \frac{3}{2} \cdot \frac{2a\sqrt{7}}{3}} = \frac{a + \frac{a\sqrt{7}}{3}}{a - a\sqrt{7}} = \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})}.
["Title: Solving a Complex Rational Equation Step-by-Step: Simplifying a Nested Fraction with Algebra", "---", "Introduction", "Working with complex rational expressions can be intimidating—especially when nested fractions involve square roots. In this SEO-optimized article, we’ll carefully break down the step-by-step simplification of a challenging fraction expression involving (a) and (\sqrt{7}). Learn how to manipulate algebraic terms, eliminate denominators, and express equivalences clearly—tools essential not only for math students but anyone mastering algebraic problem-solving.", "---", "### The Original Expression", "Start with:", "[\n\frac{a + \frac{1}{2} \cdot \frac{2a\sqrt{7}}{3}}{a - \frac{3}{2} \cdot \frac{2a\sqrt{7}}{3}} = \frac{a + \frac{a\sqrt{7}}{3}}{a - a\sqrt{7}}\n]", "This form reveals the key steps: simplify inner fractions first, then combine like terms.", "---", "### Step 1: Simplify the Numerator", "The numerator is:", "[\na + \frac{1}{2} \cdot \frac{2a\sqrt{7}}{3}\n]", "Multiply the constants:", "[\n\frac{1}{2} \cdot \frac{2a\sqrt{7}}{3} = \frac{2a\sqrt{7}}{6} = \frac{a\sqrt{7}}{3}\n]", "So the numerator becomes:", "[\na + \frac{a\sqrt{7}}{3}\n]", "This matches the simplified numerator in the next expression.", "---", "### Step 2: Simplify the Denominator", "Now simplify:", "[\na - \frac{3}{2} \cdot \frac{2a\sqrt{7}}{3}\n]", "Multiply inside:", "[\n\frac{3}{2} \cdot \frac{2a\sqrt{7}}{3} = \frac{3 \cdot 2a\sqrt{7}}{2 \cdot 3} = a\sqrt{7}\n]", "Thus, the denominator becomes:", "[\na - a\sqrt{7} = a(1 - \sqrt{7})\n]", "---", "### Step 3: Combine Simplified Numerator and Denominator", "Now the full expression simplifies to:", "[\n\frac{a + \frac{a\sqrt{7}}{3}}{a - a\sqrt{7}} = \frac{a\left(1 + \frac{\sqrt{7}}{3}\right)}{a(1 - \sqrt{7})}\n]", "Cancel (a) (assuming (a <br/>\neq 0)):", "[\n\frac{1 + \frac{\sqrt{7}}{3}}{1 - \sqrt{7}}\n]", "---", "### Step 4: Express Numerator with Common Denominator", "Write the numerator with a unified denominator:", "[\n1 + \frac{\sqrt{7}}{3} = \frac{3}{3} + \frac{\sqrt{7}}{3} = \frac{3 + \sqrt{7}}{3}\n]", "So the expression becomes:", "[\n\frac{\frac{3 + \sqrt{7}}{3}}{1 - \sqrt{7}} = \frac{3 + \sqrt{7}}{3(1 - \sqrt{7})}\n]", "---", "### Final Result", "We’ve shown how:", "[\n\frac{a + \frac{1}{2} \cdot \frac{2a\sqrt{7}}{3}}{a - \frac{3}{2} \cdot \frac{2a\sqrt{7}}{3}} = \frac{3 + \sqrt{7}}{3(1 - \sqrt{7})}\n]", "This complete simplification demonstrates key algebraic techniques:", "- Breaking down nested fractions\n- Distributing and simplifying products\n- Combining like terms with irrational numbers\n- Expressing rational expressions in clean canonical form", "These steps improve not only correctness, but also clarity—critical when explaining or optimizing algebraic content for search engines and readers alike.", "---", "### SEO Optimization Notes", "- This article targets high-intent academic searches like “solving complex rational equations,” “simplify nested fractions,” or “how to simplify expressions with square roots.”\n- The structured, step-by-step approach boosts readability and supports keyword-rich content.\n- Emphasis on simplification and cancellation highlights essential algebraic strategies trusted by educators and learners.\n- Internal linking opportunities exist (e.g., tutorials on fraction simplification, rational expressions, or conjugate methods).", "---", "Key Takeaways", "- Always simplify inner multiplication before combining.\n- Watch out for common factors—canceling (a) is valid when (a <br/>\ne 0).\n- Rational expressions with radicals often simplify neatly by expressing numerators over a common denominator.\n- Clear, step-by-step notation improves SEO and user engagement.", "---", "Further Reading", "- Full Guide to Simplifying Rational Expressions\n- Rationalizing Denominators with Square Roots\n- Mastering Nested Fractions in Algebra", "---", "Keywords: algebraic simplification, rational expressions, complex fraction simplification, solving equations with radicals, step-by-step algebra, solve nested fractions, Elementary Algebra, Rational Equation Simplification, √7 algebra, fraction expansion, conjugate rationalization.", "---", "Unlock algebraic mastery by simplifying complex expressions—one step at a time."]
