Area = √[21(21 - 13)(21 - 14)(21 - 15)] = √[21 * 8 * 7 * 6].

["# Solving Area Expression: Area = √[21(21 - 13)(21 - 14)(21 - 15)]", "Calculating areas can be both a mathematical and elegant endeavor, especially when recognizing hidden patterns or simplifications. One intriguing problem involves evaluating the area expression:", "[\n\ ext{Area} = \sqrt{21(21 - 13)(21 - 14)(21 - 15)}\n]", "### Simplifying the Expression Inside the Square Root", "Start by computing each term inside the parentheses:", "- (21 - 13 = 8)\n- (21 - 14 = 7)\n- (21 - 15 = 6)", "Substituting these values gives:", "[\n\ ext{Area} = \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]", "Now simplify the multiplication step-by-step.", "First, group the terms strategically:", "[\n(21 \ imes 7) \ imes (8 \ imes 6)\n]", "Calculate each pair:", "- (21 \ imes 7 = 147)\n- (8 \ imes 6 = 48)", "Now multiply:", "[\n147 \ imes 48\n]", "Compute (147 \ imes 48):", "- (147 \ imes 40 = 5880)\n- (147 \ imes 8 = 1176)\n- Total: (5880 + 1176 = 7056)", "So,", "[\n\ ext{Area} = \sqrt{7056}\n]", "### Evaluating the Square Root", "Next, compute:", "[\n\sqrt{7056}\n]", "Note that (84^2 = 7056), since:", "- (80^2 = 6400)\n- (4^2 = 16)\n- Cross term: (2 \ imes 80 \ imes 4 = 640)", "Adding: (6400 + 640 + 16 = 7056)", "Thus,", "[\n\sqrt{7056} = 84\n]", "### Why This Calculation Matters: A Quick Insight", "This problem demonstrates a common algebraic technique: simplifying products under square roots using arithmetic identities and factorization. Recognizing that:", "[\nn(n - a)(n - b)(n - c)\n]", "often leads to simplifiable forms involving perfect squares, especially when differences of integers multiply neatly.", "Moreover, this exact expression arises naturally in geometric contexts, such as computing areas of quadrilaterals or using Heron’s formula in related problems.", "### Practical Takeaway", "To compute such area expressions efficiently:", "1. Reduce differences inside the parentheses.\n2. Group terms to form recognizable squares or products.\n3. Simplify step-by-step before applying square roots.\n4. Always verify your result by cross-checking with direct multiplication or known square values.", "Understanding such algebraic simplifications builds a strong foundation for tackling complex geometry and optimization problems with confidence.", "---", "Conclusion:\nThe area derived from the expression √[21(21 − 13)(21 − 14)(21 − 15)] is exactly 84, derived through careful simplification and verification. Use this method to unlock elegant solutions in geometry and advanced algebra."]