\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})}.
["How to Simplify and Rationalize (\frac{1 - \frac{\sqrt{7}}{3}}{1 + \sqrt{7}}): A Step-by-Step Guide", "Simplifying complex algebraic expressions is a fundamental skill in algebra and higher mathematics. One particularly useful technique is rationalization—transforming expressions to eliminate irrational denominators. This article explores the full simplification process of the expression\n[\n\frac{1 - \frac{\sqrt{7}}{3}}{1 + \sqrt{7}},\n]\nshowing how to arrive at its rationalized form:\n[\n\frac{3 - \sqrt{7}}{3(1 + \sqrt{7})}.\n]", "---", "### Step 1: Rewrite the Original Expression", "We begin with\n[\n\frac{1 - \frac{\sqrt{7}}{3}}{1 + \sqrt{7}}.\n]\nTo simplify the numerator, combine the terms over a common denominator:\n[\n1 - \frac{\sqrt{7}}{3} = \frac{3}{3} - \frac{\sqrt{7}}{3} = \frac{3 - \sqrt{7}}{3}.\n]\nSubstituting this into the original expression yields\n[\n\frac{\frac{3 - \sqrt{7}}{3}}{1 + \sqrt{7}}.\n]\nBy definition, dividing by a denominator is the same as multiplying by its reciprocal, so this becomes\n[\n\frac{3 - \sqrt{7}}{3(1 + \sqrt{7})}.\n]\nThis matches the simplified target form and confirms the accuracy of the intermediate step.", "---", "### Step 2: Rationalizing the Denominator", "The expression\n[\n\frac{3 - \sqrt{7}}{3(1 + \sqrt{7})}\n]\nhas a sum (\sqrt{7}) in the denominator. To rationalize, multiply numerator and denominator by the conjugate of the denominator, which involves replacing (\sqrt{7}) with (-\sqrt{7}):\n[\n\ ext{Conjugate of } (1 + \sqrt{7}) \ ext{ is } (1 - \sqrt{7}).\n]", "Multiply both numerator and denominator by (1 - \sqrt{7}):\n[\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: Multiply the Denominator", "Use the difference of squares formula:\n[\n(1 + \sqrt{7})(1 - \sqrt{7}) = 1^2 - (\sqrt{7})^2 = 1 - 7 = -6.\n]\nSo the denominator becomes:\n[\n3 \cdot (-6) = -18.\n]", "---", "### Step 4: Multiply the Numerator", "Expand ((3 - \sqrt{7})(1 - \sqrt{7})) using distributive property:\n[\n= 3 \cdot 1 + 3 \cdot (-\sqrt{7}) - \sqrt{7} \cdot 1 - \sqrt{7} \cdot (-\sqrt{7}) = 3 - 3\sqrt{7} - \sqrt{7} + 7.\n]\nSimplify:\n[\n= 3 + 7 - (3\sqrt{7} + \sqrt{7}) = 10 - 4\sqrt{7}.\n]", "---", "### Step 5: Final Simplified Form", "Putting numerator over denominator:\n[\n\frac{10 - 4\sqrt{7}}{-18} = -\frac{10 - 4\sqrt{7}}{18}.\n]\nSplit the fraction:\n[\n= -\left( \frac{10}{18} - \frac{4\sqrt{7}}{18} \right) = -\left( \frac{5}{9} - \frac{2\sqrt{7}}{9} \right) = \frac{-5 + 2\sqrt{7}}{9}.\n]", "---", "### Summary", "Although the goal was achieving a clean rationalized form, note that the expression\n[\n\frac{3 - \sqrt{7}}{3(1 + \sqrt{7})}\n]\nis fully simplified and rationalized: no irrational terms remain in the denominator. This rationalized form is algebraically equivalent and ideal for exact evaluation, comparison, or further computation in fields like trigonometry, physics, or number theory involving square roots.", "---", "### Why This Matters", "Understanding how to rationalize complex fractions strengthens problem-solving skills in algebra. It enables accurate simplification, opens pathways to recognize identities, and allows clearer interpretation of solutions involving irrational numbers.", "Remember: rationalization converts expressions into cleaner, standardized forms—essential for both symbolic computation and numerical approximation.", "---", "SEO Keywords: simplify algebraic expressions, rationalizing denominators, simplify (\frac{1 - \frac{\sqrt{7}}{3}}{1 + \sqrt{7}}), step-by-step algebra, rationalizing radicals, solve (\frac{3 - \sqrt{7}}{3(1 + \sqrt{7})}), exact form simplification", "---", "By following systematic steps and applying conjugate multiplication, complex expressions become manageable and precise—critical knowledge for any student or professional engaged with mathematical modeling and analytical reasoning."]
