\frac{(3 + \sqrt{7})(1 + \sqrt{7})}{3(1 - 7)} = \frac{3(1) + 3\sqrt{7} + \sqrt{7} + 7}{3(-6)} = \frac{10 + 4\sqrt{7}}{-18} = -\frac{10 + 4\sqrt{7}}{18} = -\frac{5 + 2\sqrt{7}}{9}.
["Simplifying a Complex Algebraic Expression: A Step-by-Step Breakdown", "Mastering algebraic simplification is essential for students, educators, and math enthusiasts alike. One illustrative example is simplifying the expression:", "$$\n\frac{(3 + \sqrt{7})(1 + \sqrt{7})}{3(1 - 7)} = \frac{3(1) + 3\sqrt{7} + \sqrt{7} + 7}{3(-6)} = \frac{10 + 4\sqrt{7}}{-18} = -\frac{10 + 4\sqrt{7}}{18} = -\frac{5 + 2\sqrt{7}}{9}\n$$", "In this article, we walk through each step clearly, emphasizing logical progress and mathematical clarity to uncover how this complex fraction simplifies beautifully into its simplest form.", "---", "### Step 1: Expand the Numerator", "Begin by multiplying the binomials in the numerator:", "$$\n(3 + \sqrt{7})(1 + \sqrt{7})\n$$", "Apply the distributive property (FOIL method):", "- First: (3 \ imes 1 = 3)\n- Outer: (3 \ imes \sqrt{7} = 3\sqrt{7})\n- Inner: (\sqrt{7} \ imes 1 = \sqrt{7})\n- Last: (\sqrt{7} \ imes \sqrt{7} = (\sqrt{7})^2 = 7)", "Adding them together:", "$$\n3 + 3\sqrt{7} + \sqrt{7} + 7 = (3 + 7) + (3\sqrt{7} + \sqrt{7}) = 10 + 4\sqrt{7}\n$$", "---", "### Step 2: Simplify the Denominator", "The denominator is (3(1 - 7)):", "$$\n1 - 7 = -6 \quad \Rightarrow \quad 3(1 - 7) = 3(-6) = -18\n$$", "---", "### Step 3: Write the Fraction in Original Form", "Substitute the expanded numerator and simplified denominator:", "$$\n\frac{10 + 4\sqrt{7}}{-18}\n$$", "---", "### Step 4: Factor and Reduce", "Factor out a common factor from the numerator and denominator:", "- Numerator: (10 + 4\sqrt{7} = 2(5 + 2\sqrt{7}))\n- Denominator: ( -18 = -6 \ imes 3 = -2 \ imes 9) → but simpler to factor 2: (-18 = -2 \ imes 9)", "However, observe the numerator can be written as:", "$$\n\frac{2(5 + 2\sqrt{7})}{-18} = -\frac{2(5 + 2\sqrt{7})}{18}\n$$", "Cancel the common factor (2):", "$$\n-\frac{5 + 2\sqrt{7}}{9}\n$$", "---", "### Conclusion: Final Simplified Form", "Thus, the original complex fraction simplifies cleanly to:", "$$\n-\frac{5 + 2\sqrt{7}}{9}\n$$", "---", "### Why Simplification Matters", "Simplifying expressions like this improves readability, reduces computational error, and enables easier integration into broader mathematical reasoning—whether solving equations, calculus problems, or advanced algebra.", "---", "Key Takeaways:", "- Always expand products carefully using distributive property.\n- Simplify coefficients first before factoring out common terms.\n- Signs matter—especially when denominators are negative.\n- Recognizing common factors accelerates reduction.", "By mastering such step-by-step techniques, anyone can confidently tackle similar algebraic challenges—turning complexity into clarity.", "---", "Keywords: algebra simplification, rational expressions, binomial multiplication, simplify radicals, fractional expressions, step-by-step algebra, solving algebraic equations, simplifying radicals, mathematical reasoning, algebra tutorials", "Meta Description:\nLearn how to simplify the expression (\frac{(3 + \sqrt{7})(1 + \sqrt{7})}{3(1 - 7)}) step-by-step. From expansion to final reduction, discover how to simplify radicals and fractions with clarity and precision."]
