rac{(\sqrt7 + \sqrt3)^2}{(\sqrt7)^2 - (\sqrt3)^2} = rac{7 + 2\sqrt21 + 3}7 - 3 = rac{10 + 2\sqrt21}4 = rac{5 + \sqrt21}2. - Parker Core Knowledge
Solving the Interesting Equation: Rac{(√7 + √3)²}{(√7)² − (√3)²} = (7 + 2√21 + 3)/(7 − 3) = (5 + √21)/2
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.
Understanding the Context
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
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Key Insights
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
Step 2: Simplify the Denominator — (√7)² − (√3)²
The denominator is a classic difference of squares:
(√7)² − (√3)²
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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.
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
Why This Equality Matters
This identity demonstrates a powerful fusion of:
- Binomial expansion for radicals
- Difference of squares formula
- Careful algebraic simplification
