x^2 = \left( \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} \right)^2 = (7 + 2\sqrt{10}) + (7 - 2\sqrt{10}) + 2\sqrt{(7 + 2\sqrt{10})(7 - 2\sqrt{10})}

["# Perfect Solution to Simplify: ( x^2 = \left( \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} \right)^2 )", "Understanding the algebraic simplification of complex square roots is crucial for mastering advanced algebra and manipulating nested radicals. One elegant example involves the expression:", "[\nx^2 = \left( \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} \right)^2\n]", "This expression may appear intimidating at first, but by breaking it down step-by-step, we uncover a path to simplify both the square and its result — all while preserving mathematical integrity.", "---", "## Why Simplify This Expression?", "Expressions with nested square roots often arise in calculus, geometry, and mathematical competitions. Simplifying such forms:", "- Reduces computational complexity\n- Reveals underlying symmetry\n- Provides exact values for algebraic identities\n- Offers insights into conjugate pairs", "In this case, we explore how squaring a sum of two conjugate square roots leads to a clean algebraic expression involving radicals and rational terms.", "---", "## Step 1: Expand the Square Expression", "Start by expanding the left-hand side using the identity ( (a + b)^2 = a^2 + b^2 + 2ab ):", "[\nx^2 = \left( \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} \right)^2 = \left( \sqrt{7 + 2\sqrt{10}} \right)^2 + \left( \sqrt{7 - 2\sqrt{10}} \right)^2 + 2 \cdot \sqrt{7 + 2\sqrt{10}} \cdot \sqrt{7 - 2\sqrt{10}}\n]", "Compute each term:", "- ( \left( \sqrt{7 + 2\sqrt{10}} \right)^2 = 7 + 2\sqrt{10} )\n- ( \left( \sqrt{7 - 2\sqrt{10}} \right)^2 = 7 - 2\sqrt{10} )\n- The product under the square root:\n [\n \sqrt{(7 + 2\sqrt{10})(7 - 2\sqrt{10})} = \sqrt{49 - (2\sqrt{10})^2} = \sqrt{49 - 40} = \sqrt{9} = 3\n ]", "So the full expansion becomes:", "[\nx^2 = (7 + 2\sqrt{10}) + (7 - 2\sqrt{10}) + 2 \cdot 3\n]", "---", "## Step 2: Combine Like Terms", "Simplify each component:", "- ( (7 + 2\sqrt{10}) + (7 - 2\sqrt{10}) = 7 + 7 + 2\sqrt{10} - 2\sqrt{10} = 14 )\n- ( 2 \cdot 3 = 6 )", "Therefore:", "[\nx^2 = 14 + 6 = 20\n]", "---", "## Final Result", "[\n\boxed{x^2 = 20} \quad \ ext{so} \quad x = \pm \sqrt{20} = \pm 2\sqrt{5}\n]", "---", "## Why This Matters", "This derivation demonstrates how conjugate radical pairs can combine neatly when squared — a technique frequently useful in:", "- Trigonometric identities involving square roots\n- Quadratic pattern recognition in nested radicals\n- Problem-solving tricks in math competitions", "By simplifying ( x^2 ), we efficiently unlock the value of ( x ) without directly computing messy nested roots.", "---", "## Key Takeaways", "| Step | Action | Reason |\n|------|--------|--------|\n| 1 | Expand using ( (a + b)^2 ) | Establishes full structure for simplification |\n| 2 | Compute each square | Eliminates radicals on outer terms |\n| 3 | Multiply product of radicals | Leverages difference of squares identity |\n| 4 | Combine rational and irrational terms | Finalizes expression concisely |", "---", "## Further Practice", "Try simplifying similar expressions such as:", "[\n\sqrt{a + 2\sqrt{b}} + \sqrt{a - 2\sqrt{b}}\n]", "or check:", "[\n\sqrt{5 + 2\sqrt{6}} + \sqrt{5 - 2\sqrt{6}}\n]", "Both follow the same elegant pattern — perfect for deepening your mastery of algebraic simplifications.", "---", "Keywords:\n( x^2 ), ( \sqrt{7 + 2\sqrt{10}} ), ( \sqrt{7 - 2\sqrt{10}} ), simplify radicals, nested square roots, algebraic identity, conjugate radicals, expand and simplify, radicals simplification, mathematical proof, algebra simplification", "---", "Meta Description:\nMaster how to simplify ( x^2 = \left( \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} \right)^2 ), learning key algebraic steps involving conjugate square roots and simplification techniques."]