Frage: Berechne das Quadrat von \( 5 + \sqrt{21} \).
["Title: How to Calculate the Square of ( 5 + \sqrt{21} ): Easy Step-by-Step Guide", "Meta Description:\nLearn how to calculate the square of ( 5 + \sqrt{21} ) using simple algebra. Maximize understanding with clear steps and practical examples.", "---", "### Preface", "Mathematics often asks us to compute expressions involving square roots — let’s simplify the common question: How do we calculate the square of ( 5 + \sqrt{21} )? This article breaks down the algebra step-by-step, making it easy to grasp and apply, whether you're a student tackling homework or someone exploring math practically.", "---", "### Understanding the Problem", "We want to compute:", "[\n(5 + \sqrt{21})^2\n]", "This expression represents the square (or product of the number with itself) of ( 5 + \sqrt{21} ). Squaring a binomial follows a well-known algebraic formula, and we’ll apply it efficiently here.", "---", "### Step 1: Apply the Binomial Square Formula", "Recall the identity:", "[\n(a + b)^2 = a^2 + 2ab + b^2\n]", "Let:\n- ( a = 5 )\n- ( b = \sqrt{21} )", "---", "### Step 2: Compute Each Term", "1. First term:\n[\na^2 = 5^2 = 25\n]", "2. Second term (cross term):\n[\n2ab = 2 \cdot 5 \cdot \sqrt{21} = 10\sqrt{21}\n]", "3. Third term:\n[\nb^2 = (\sqrt{21})^2 = 21\n]", "---", "### Step 3: Add All Terms Together", "Combine all parts:", "[\n(5 + \sqrt{21})^2 = 25 + 10\sqrt{21} + 21\n]", "Simplify by adding constants:", "[\n25 + 21 = 46\n]", "So,", "[\n(5 + \sqrt{21})^2 = 46 + 10\sqrt{21}\n]", "---", "### Final Answer", "[\n\boxed{(5 + \sqrt{21})^2 = 46 + 10\sqrt{21}}\n]", "---", "### Why This Matters", "Understanding how to square binomials involving radicals enhances problem-solving in algebra, geometry, and applied sciences. The result, ( 46 + 10\sqrt{21} ), is an exact expression — precise and useful in deeper mathematical explorations.", "---", "### Summary", "- Use ( (a + b)^2 = a^2 + 2ab + b^2 ).\n- Substitute ( a = 5 ), ( b = \sqrt{21} ).\n- Compute each term carefully.\n- Combine constants and radicals properly.", "---", "Related Keywords for SEO:", "- Square of binomial ( 5 + \sqrt{21} )\n- Calculate ( (5 + \sqrt{21})^2 )\n- Simplify square of radical expression\n- Algebra realization ( (a + b)^2 )\n- Exact form square root expressions", "---", "Ideal for students, educators, and math enthusiasts — this guide demystifies an essential algebraic computation with clarity and practical insight."]
