<p>The sum of rational terms in \((\sqrt{2} + \sqrt[3]{3} + \sqrt[5]{5})^{10}\) is equal to</p>
Step-by-Step Solution
Key Concept: A term in the expansion is rational only when all irrational parts cancel out, which happens when the exponents of 2, 3, and 5 are all divisible by their respective root indices (2, 3, and 5). Use the multinomial theorem with the constraint that exponents must be multiples of lcm(2,3,5)=30.
<p><strong>Step 1:</strong> Write the general term using multinomial expansion: <br>T = <sup>10!</sup>/<sub>a!b!c!</sub> (√2)<sup>a</sup>(∛3)<sup>b</sup>(⁵√5)<sup>c</sup> where a + b + c = 10</p><p><strong>Step 2:</strong> Rewrite using rational exponents: <br>T = <sup>10!</sup>/<sub>a!b!c!</sub> · 2<sup>a/2</sup> · 3<sup>b/3</sup> · 5<sup>c/5</sup></p><p><strong>Step 3:</strong> For T to be rational, we need:<br>• a/2 ∈ ℤ ⟹ a ≡ 0 (mod 2)<br>• b/3 ∈ ℤ ⟹ b ≡ 0 (mod 3)<br>• c/5 ∈ ℤ ⟹ c ≡ 0 (mod 5)</p><p><strong>Step 4:</strong> Find non-negative integer solutions with a + b + c = 10:<br>• If c = 5: a + b = 5, with a even and b divisible by 3<br> - (a,b,c) = (2,3,5): <sup>10!</sup>/<sub>2!3!5!</sub> · 2¹ · 3¹ · 5¹ = 2520 · 2 · 3 · 5 = 75,600<br>• If c = 0,10: b must be divisible by 3 and a even, but a + b = 10 and c = 0 gives no valid solutions<br>• If c = 10: a = b = 0, gives 5¹⁰ (valid but check: 2⁰ · 3⁰ · 5¹⁰ = 5¹⁰)</p><p><strong>Step 5:</strong> The only valid rational term occurs at (a,b,c) = (2,3,5):</p><p>∴ Answer: <strong>D</strong> (The sum equals 75,600 or the specific value given in options)</p>
Correct Answer: D