<p>Find the term independent of \(x\) in the expansion of \(\left(2x^2 - \dfrac{3}{x^3}\right)^{25}\).</p>
Step-by-Step Solution
Key Concept: The general term in the binomial expansion is T_{r+1} = C(25,r)(2x²)^{25-r}(-3/x³)^r. For the term independent of x, the power of x must equal zero: 2(25-r) - 3r = 0.
<p><strong>Step 1:</strong> Write the general term of the binomial expansion.</p><p>T_{r+1} = C(25,r)(2x²)^{25-r}·(-3/x³)^r</p><p><strong>Step 2:</strong> Simplify to find the power of x.</p><p>T_{r+1} = C(25,r)·2^{25-r}·(-3)^r·x^{2(25-r)-3r}</p><p>T_{r+1} = C(25,r)·2^{25-r}·(-3)^r·x^{50-2r-3r}</p><p>T_{r+1} = C(25,r)·2^{25-r}·(-3)^r·x^{50-5r}</p><p><strong>Step 3:</strong> Set the power of x equal to zero for the independent term.</p><p>50 - 5r = 0</p><p>r = 10</p><p><strong>Step 4:</strong> Calculate the term independent of x.</p><p>T_{11} = C(25,10)·2^{15}·(-3)^{10}</p><p>T_{11} = C(25,10)·2^{15}·3^{10}</p><p>C(25,10) = 3,268,760</p><p>∴ Answer: <strong>3,268,760 × 2^{15} × 3^{10}</strong> or <strong>C(25,10)·2^{15}·3^{10}</strong></p>
Correct Answer: 3