<p>Find the 7th term of \(\left(3x^2 - \dfrac{1}{3}\right)^{10}\).</p>
Step-by-Step Solution
Key Concept: In the binomial expansion of (a+b)^n, the rth term is T_r = C(n,r-1)·a^(n-r+1)·b^(r-1). For the 7th term, use r=7 with a=3x², b=-1/3, and n=10.
<p><strong>Step 1:</strong> Identify parameters: n=10, a=3x², b=-1/3, and we need the 7th term.</p><p><strong>Step 2:</strong> Use general term formula T_r = C(n,r-1)·a^(n-r+1)·b^(r-1).</p><p>For 7th term: T_7 = C(10,6)·(3x²)^(10-6)·(-1/3)^(6-1)</p><p><strong>Step 3:</strong> Calculate C(10,6) = 210</p><p><strong>Step 4:</strong> Simplify powers: (3x²)^4 = 81x^8 and (-1/3)^5 = -1/243</p><p><strong>Step 5:</strong> Combine: T_7 = 210 · 81x^8 · (-1/243)</p><p><strong>Step 6:</strong> T_7 = 210 · 81 · (-1) · x^8 / 243 = -210 · 81x^8 / 243 = -210x^8/3 = -70x^8</p><p>∴ Answer: <strong>-70x^8</strong></p>
Correct Answer: -70