Solving the Interesting Equation: Rac{(√7 + √3)²}{(√7)² − (√3)²} = (7 + 2√21 + 3)/(7 − 3) = (5 + √21)/2

Mathematics often hides elegant truths beneath layers of symbols and operations. One such intriguing relation involves radical expressions: rac{(√7 + √3)²}{(√7)² − (√3)²} = (5 + √21)/2

In this article, we’ll break down this identity step-by-step, clarify implicit steps, and explore why this result—combining binomial expansion, algebraic simplification, and the difference of squares—is both elegant and instructive.

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Step 1: Expand the Numerator — (√7 + √3)²

The expression begins with the numerator: (√7 + √3)²

Using the algebraic identity: (a + b)² = a² + 2ab + b² we expand: (√7 + √3)² = (√7)² + 2(√7)(√3) + (√3)²

Calculate each term:

• (√7)² = 7
• (√3)² = 3
• 2(√7)(√3) = 2√(7·3) = 2√21

Thus, (√7 + √3)² = 7 + 2√21 + 3 = 10 + 2√21

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Step 2: Simplify the Denominator — (√7)² − (√3)²

The denominator is a classic difference of squares: (√7)² − (√3)²

Apply the identity: a² − b² = (a − b)(a + b) but here we can directly simplify: (√7)² = 7, (√3)² = 3 ⇒ 7 − 3 = 4

So, the denominator becomes 4.

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Step 3: Combine Numerator and Denominator

Now substitute the simplified forms back into the original fraction: rac{(√7 + √3)²}{(√7)² − (√3)²} = (10 + 2√21) / 4

Factor numerator: = [2(5 + √21)] / 4 = (5 + √21) / 2

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Why This Equality Matters

This identity demonstrates a powerful fusion of:

• Binomial expansion for radicals
• Difference of squares formula
• Careful algebraic simplification

It confirms that expansion and simplification at each step preserves equality — a foundational principle in algebra. Understanding such transformations strengthens problem-solving skills and clarity in advanced mathematics, including algebra, calculus, and engineering applications.

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Final Result

Therefore, rac{(√7 + √3)²}{(√7)² − (√3)²} = (5 + √21) / 2

A concise yet profound result emerging from thoughtful step-by-step reasoning.

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Whether you’re studying for exams or exploring mathematical beauty, mastering expressions like this sharpens your analytical mindset and appreciation for mathematical harmony.

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