Question: Compute the square of $ \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} $.
["Title: Compute the Square of $ \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} $ – Step-by-Step Explanation", "When you encounter a math problem like “Compute the square of $ \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} $”, it’s a classic example of simplifying a complicated expression using algebraic identities and square root manipulation. This article guides you through the step-by-step process to find the square of this expression, revealing its elegant and simpler algebraic form.", "---", "### Understanding the Problem", "We are asked to compute:", "[\n\left( \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} \right)^2\n]", "At first glance, directly squaring this expression seems messy due to the nested square roots. However, by applying the identity:", "[\n(a + b)^2 = a^2 + 2ab + b^2\n]", "we can simplify the computation significantly—provided we first determine $ a^2 $, $ b^2 $, and simplify $ 2ab $ accurately.", "---", "### Step 1: Define the Expression", "Let\n[\nx = \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}}\n]", "We want to compute $ x^2 $.", "---", "### Step 2: Expand the Square", "Using the identity $ (a + b)^2 = a^2 + 2ab + b^2 $, we compute:", "[\nx^2 = \left( \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} \right)^2 = (7 + 2\sqrt{10}) + (7 - 2\sqrt{10}) + 2 \cdot \sqrt{7 + 2\sqrt{10}} \cdot \sqrt{7 - 2\sqrt{10}}\n]", "Simplify the terms inside the parentheses:", "[\n= 7 + 2\sqrt{10} + 7 - 2\sqrt{10} + 2 \cdot \sqrt{(7 + 2\sqrt{10})(7 - 2\sqrt{10})}\n]", "The $ 2\sqrt{10} $ terms cancel:", "[\n= 14 + 2 \cdot \sqrt{(7 + 2\sqrt{10})(7 - 2\sqrt{10})}\n]", "---", "### Step 3: Simplify the Product Under the Square Root", "We now simplify:", "[\n(7 + 2\sqrt{10})(7 - 2\sqrt{10})\n]", "This is a difference of squares:\n[\na^2 - b^2 = (7)^2 - (2\sqrt{10})^2 = 49 - 4 \cdot 10 = 49 - 40 = 9\n]", "So,", "[\n\sqrt{(7 + 2\sqrt{10})(7 - 2\sqrt{10})} = \sqrt{9} = 3\n]", "---", "### Step 4: Complete the Calculation", "Substitute back into the expression:", "[\nx^2 = 14 + 2 \cdot 3 = 14 + 6 = 20\n]", "---", "### Conclusion", "The square of\n[\n\sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}}\n]\nis simply:", "[\n\boxed{20}\n]", "This elegant result demonstrates how algebraic identities can transform complex radical expressions into simple numerical values—powerful tools in algebra and problem-solving. Next time you face a similar expression, apply these steps methodically and discover the hidden simplicity beneath.", "---", "Keywords: square of square roots, simplify nested radicals, algebraic identity, $ (\sqrt{a} + \sqrt{b})^2 $, $ \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} $, mathematical simplification, radical expressions, algebra problem-solving", "Meta Description: Learn how to compute the square of $ \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} $ step-by-step using algebraic identities. Discover the elegant result $ \boxed{20} $ and master simplification techniques for complex radicals."]
