Algebraic simplification (compound fraction): If a = p/(p + q) and b = q/(p − q), evaluate ab / (a + b) in simplest form.
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Apq / (p^2 + q^2)
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B(p^2 + q^2) / pq
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Cp / (p + q)
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D[p / (p + q)]^2
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Eq / (p − q)
Answer
Correct Answer: pq / (p^2 + q^2)
Explanation
Introduction / Context:This problem requires careful manipulation of rational expressions. By putting a and b over a common denominator, we can form a + b and ab explicitly and then simplify the compound fraction ab/(a + b). The key move is recognizing that identical factors cancel cleanly.
Given Data / Assumptions:
- a = p/(p + q)
- b = q/(p − q)
- p ≠ ±q (to avoid zero denominators)
Concept / Approach:Compute numerator and denominator separately: ab and a + b. Use the identity (p + q)(p − q) = p^2 − q^2 and simplify. The cancellation of (p^2 − q^2) between numerator and denominator will leave a compact expression in p and q.
Step-by-Step Solution:
ab = [p/(p + q)] * [q/(p − q)] = pq / (p^2 − q^2)a + b = p/(p + q) + q/(p − q) = [p(p − q) + q(p + q)] / (p^2 − q^2)Simplify numerator: p^2 − pq + qp + q^2 = p^2 + q^2Thus ab/(a + b) = [pq/(p^2 − q^2)] / [(p^2 + q^2)/(p^2 − q^2)] = pq/(p^2 + q^2)Verification / Alternative check:Test with concrete values (e.g., p = 3, q = 1) to ensure the simplified expression matches the numeric computation of ab/(a + b). It does.
Why Other Options Are Wrong:
- Other options either invert the result or drop necessary terms, failing to account for the combination in the denominator.
Common Pitfalls:Forgetting to combine like terms p(−q) + q(p) = 0 (they cancel), leaving p^2 + q^2. Also, do not try to add numerators without establishing the common denominator first.
Final Answer:pq / (p^2 + q^2)