If √(43 − 24√3) = a + b√3, then what is the value of (3a + 5b)?

Let √(43 − 24√3) = a + b√3.

Square both sides:

43 − 24√3 = (a + b√3)² = a² + 3b² + 2ab√3.

Equate the rational and the √3 parts to get:

• a² + 3b² = 43

• 2ab = −24, so ab = −12

Find integer pairs with ab = −12 and check the first equation. The relevant pairs are (a, b) = (4, −3) and (a, b) = (−4, 3); both satisfy a² + 3b² = 43.

Decide which pair matches the principal (nonnegative) square root by evaluating a + b√3:

• For a = 4, b = −3: a + b√3 = 4 − 3√3 ≈ −1.196 (negative), so this is not the principal square root.

• For a = −4, b = 3: a + b√3 = −4 + 3√3 ≈ 1.196 (positive), so this is the principal square root value.

Therefore choose a = −4 and b = 3. Now compute 3a + 5b = 3(−4) + 5(3) = −12 + 15 = 3.

Answer: 3.