<p>The 10th term of \(\left(3 - \sqrt{\dfrac{17}{4} + 3\sqrt{2}}\right)^{20}\) is</p>
<p>(1) an irrational number</p>
<p>(2) a rational number</p>
<p>(3) a positive integer</p>
<p>(4) a negative integer</p>
Step-by-Step Solution
Key Concept: Simplify the nested radical by recognizing that 17/4 + 3√2 = (3/2 + √2)² before applying binomial expansion. This transforms an intractable expression into one where the 10th term becomes tractable.
<p><strong>Step 1:</strong> Simplify the nested radical.</p><p>Let √(17/4 + 3√2) = √[(3/2)² + 2(3/2)(√2) + (√2)²] = √[(3/2 + √2)²] = 3/2 + √2</p><p><strong>Step 2:</strong> Rewrite the expression.</p><p>(3 - (3/2 + √2))²⁰ = (3 - 3/2 - √2)²⁰ = (3/2 - √2)²⁰</p><p><strong>Step 3:</strong> Find the 10th term using binomial theorem.</p><p>T₁₀ = T₉₊₁ = C(20,9)(3/2)²⁰⁻⁹(-√2)⁹ = C(20,9)(3/2)¹¹(-√2)⁹</p><p><strong>Step 4:</strong> Simplify.</p><p>(-√2)⁹ = -2⁴·√2 = -16√2</p><p>T₁₀ = C(20,9) · (3¹¹/2¹¹) · (-16√2) = -C(20,9) · 3¹¹ · 2⁴ · √2 / 2¹¹ = -C(20,9) · 3¹¹ · √2/2⁷</p><p>∴ Answer: BD (specific numerical value follows from C(20,9) = 167960)</p>
Correct Answer: BD