|ec{OA}| = \sqrt{(-1 + \sqrt{7})^2 + (1 + \sqrt{7})^2} = \sqrt{(1 - 2\sqrt{7} + 7) + (1 + 2\sqrt{7} + 7)} = \sqrt{16 + 8} = \sqrt{24} = 2\sqrt{6}

["Understanding the Expression |OA| = √[(-1 + √7)² + (1 + √7)²] and Its Simplification", "Mathematics often reveals elegant simplifications beneath complex-looking expressions, and the calculation of the magnitude |OA| serves as a prime example. In this article, we explore how the expression |OA| = √[(-1 + √7)² + (1 + √7)²] simplifies beautifully to |OA| = 2√6 through thoughtful algebraic manipulation.", "---", "### What is |OA|?", "In geometry, |OA| typically denotes the magnitude (or length) of a vector from the origin O to point A, assuming A has coordinates tied to the given expression. Here, instead of simple numeric coordinates, we work with algebraic terms involving √7, inviting simplification via properties of square roots and perfect squares.", "---", "### A Step-by-Step Breakdown", "Start with the original expression:\n[\n|OA| = \sqrt{(-1 + \sqrt{7})^2 + (1 + \sqrt{7})^2}\n]", "First, expand each squared term carefully.", "Step 1: Expand (-1 + √7)²\n[\n(-1 + \sqrt{7})^2 = (-1)^2 + 2(-1)(\sqrt{7}) + (\sqrt{7})^2 = 1 - 2\sqrt{7} + 7 = 8 - 2\sqrt{7}\n]", "Step 2: Expand (1 + √7)²\n[\n(1 + \sqrt{7})^2 = 1^2 + 2(1)(\sqrt{7}) + (\sqrt{7})^2 = 1 + 2\sqrt{7} + 7 = 8 + 2\sqrt{7}\n]", "Step 3: Add the two expanded expressions\n[\n(8 - 2\sqrt{7}) + (8 + 2\sqrt{7}) = 8 + 8 - 2\sqrt{7} + 2\sqrt{7} = 16\n]", "Step 4: Take the square root of the sum\n[\n|OA| = \sqrt{16} = 4\n]", "Wait — this result contradicts the claim we’re aiming for (√24 = 2√6). Let’s reevaluate.", "---", "### Correction: Full Simplification Combining Like Terms", "Actually, let’s revisit the entire expression carefully and ensure no sign errors:", "We start again:\n[\n|OA| = \sqrt{(-1 + \sqrt{7})^2 + (1 + \sqrt{7})^2}\n]", "Expand both squares as before:", "- ((-1 + \sqrt{7})^2 = 1 - 2\sqrt{7} + 7 = 8 - 2\sqrt{7})\n- ((1 + \sqrt{7})^2 = 1 + 2\sqrt{7} + 7 = 8 + 2\sqrt{7})", "Add the two:", "[\n(8 - 2\sqrt{7}) + (8 + 2\sqrt{7}) = 16\n]", "Hence,", "[\n|OA| = \sqrt{16} = 4\n]", "This confirms our earlier result — but this contradicts √24 = 2√6 ≈ 4.899.", "---", "### Clarifying the Purpose of the Expression", "While the direct computation gives |OA| = 4, the expression’s intended value appears not just as a computation but as a mathematical reveal — illustrating how irrational components involving √7 combine algebraically to simplify to a clean irrational number or simplified radical.", "However, observe:", "[\n\sqrt{(-1 + \sqrt{7})^2 + (1 + \sqrt{7})^2}\n]", "Can be rewritten as:", "[\n\sqrt{( (\sqrt{7} - 1)^2 + (\sqrt{7} + 1)^2 )} = \sqrt{ 2(\sqrt{7}^2 + 1^2) } = \sqrt{2(7 + 1)} = \sqrt{16} = 4\n]", "This uses the algebraic identity:\n[\n(a - b)^2 + (a + b)^2 = 2a^2 + 2b^2\n]", "So with ( a = \sqrt{7}, b = 1 ), we get:", "[\n|OA| = \sqrt{2( (\sqrt{7})^2 + 1^2 )} = \sqrt{2(7 + 1)} = \sqrt{16} = 4\n]", "Thus, |OA| = 4, a precise rational number — yet this highlights the power of algebraic structure.", "---", "### But where does 2√6 come from?", "There appears to be a misprint or confusion in the target value √24 = 2√6. Let’s compute √24:", "[\n\sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6} \approx 4.899\n]", "But from above, |OA| = 4 — a clean rational number.", "So either the expression simplifies to 4, or the intended value was symbolic.", "Alternatively, the expression:", "[\n\sqrt{(-1 + \sqrt{7})^2 + (1 + \sqrt{7})^2}\n]", "can be seen as a norm in a 2D vector space over ℝ, where components are real-valued functions involving √7. In geometry or linear algebra contexts, simplifying such expressions reveals symmetry or invariance — here, orthogonality (via cross cancellation in expansion).", "But unless additional geometric interpretation is provided (e.g., points at (-1,1) and (1,√7) relative to origin), |OA| remains purely algebraic.", "---", "### Summary — The True Takeaway", "- Direct algebraic simplification of\n [\n |OA| = \sqrt{(-1 + \sqrt{7})^2 + (1 + \sqrt{7})^2} = 4\n ]\n confirms a rational outcome through cancellation in mixed terms.", "- While 2√6 ≈ 4.899, it does not match |OA| = 4.", "- However, the expression beautifully demonstrates:", "[\n (a - b)^2 + (a + b)^2 = 2a^2 + 2b^2\n ]", "leading to elegant simplification.", "---", "### Final Note", "When encountering expressions like this, carefully expand and combine like terms. Even if a curious target number appears, the algebraic story often reveals deeper structure — in this case, a clean demonstration of radical expansion and symmetry, not an approximation.", "For further exploration, consider applying this identity to other binomials and observe how cross-terms interact.", "---", "Key Takeaway:\nThe magnitude expression simplifies accurately to |OA| = 4, showcasing how seemingly complex square roots collapse into rational numbers through algebraic identity — a reminder of math’s hidden harmony.", "---", "Keywords: |OA|, simplify radical, √7 algebra, vector magnitude, algebraic simplification, exponential expressions, rational roots, mathematical identity", "Meta Description:\nExplore the simplification of the expression |OA| = √[(-1 + √7)² + (1 + √7)²]. Discover step-by-step algebra, arithmetic truth (|OA| = 4), and insights into radical identities—crafting clarity from complexity."]