Rationalize and simplify the surd expression: If (5 + 2√3) / (7 + 4√3) = a + b√3, determine the exact values of a and b.

Introduction / Context:
Converting a fraction with surds into the form a + b√3 requires rationalizing the denominator. By multiplying numerator and denominator by the conjugate of the denominator, we eliminate the surd from the denominator and then collect rational and irrational parts separately.

Given Data / Assumptions:

• Original expression: (5 + 2√3) / (7 + 4√3).
• Target form: a + b√3 where a and b are rational numbers.
• Conjugate of (7 + 4√3) is (7 − 4√3).

Concept / Approach:
Multiply numerator and denominator by the conjugate (7 − 4√3). Use (p + q√3)(p − q√3) = p^2 − 3q^2 to remove the surd from the denominator. Then expand the numerator and separate terms with and without √3 to identify a and b.

Step-by-Step Solution:

Verification / Alternative check:
Because the rationalized denominator equals 1, the product in the numerator is already the final simplified value. Quick numerical check: evaluate both sides approximately to confirm consistency.

Why Other Options Are Wrong:
The signs of a and b are determined by the cross terms (−20√3 + 14√3 = −6√3). Any option with b positive or wrong a-value ignores this cancellation.

Common Pitfalls:
Forgetting that (√3)^2 = 3 when expanding, or using the wrong conjugate sign leading to an incorrect denominator.

Final Answer:
a = 11, b = -6