<p>Find the term independent of <em>x</em> in the expansion of \(\left(3x - \dfrac{1}{x}\right)^{20}\). If this term is <em>A</em>, then find the term independent of <em>x</em> in the expansion of \(\left(x + \dfrac{\sqrt[9]{3^{10}}}{x}\right)^{18}\).</p>
Step-by-Step Solution
Key Concept: For a term to be independent of x in a binomial expansion, the power of x must equal zero. Use the general term formula T_{r+1} and set the exponent of x to zero, then solve for r. The second part requires recognizing that ∛[9]{3^10} = 3^{10/9} and applying the same method.
<p><strong>Step 1: Find the independent term in (3x - 1/x)^20</strong></p><p>General term: T_{r+1} = C(20,r) · (3x)^{20-r} · (-1/x)^r = C(20,r) · 3^{20-r} · (-1)^r · x^{20-r-r}</p><p>For independence from x: 20 - 2r = 0 → r = 10</p><p>T_{11} = C(20,10) · 3^{10} · (-1)^{10} = C(20,10) · 3^{10}</p><p>A = 184756 · 59049 = 10,916,133,644 [or express as C(20,10) · 3^{10}]</p><p><strong>Step 2: Find independent term in (x + ∛[9]{3^{10}}/x)^{18}</strong></p><p>Simplify: ∛[9]{3^{10}} = 3^{10/9}</p><p>General term: T_{r+1} = C(18,r) · x^{18-r} · (3^{10/9})^r · x^{-r} = C(18,r) · 3^{10r/9} · x^{18-2r}</p><p>For independence: 18 - 2r = 0 → r = 9</p><p>T_{10} = C(18,9) · 3^{90/9} = C(18,9) · 3^{10}</p><p>= 48620 · 59049 = 2,869,685,580 [or C(18,9) · 3^{10}]</p><p>∴ Answer: B</p>
Correct Answer: B