<p>If the 6<sup>th</sup> term in the expansion of \(\left(\dfrac{1}{x^{8/3}} + x^2 \log_{10} x\right)^8\) is 5600, then \(x\) equals</p>
Step-by-Step Solution
Key Concept: The 6th term in binomial expansion uses T_{r+1} = C(n,r)a^{n-r}b^r where r=5. After substituting and simplifying the powers of x, equate the resulting expression to 5600 to find x.
<p><strong>Step 1:</strong> Identify the 6th term using T₆ = T₅₊₁ = C(8,5)·(1/x^{8/3})^{8-5}·(x² log₁₀ x)^5</p><p><strong>Step 2:</strong> Simplify: T₆ = C(8,5)·x^{-8}·x^{10}·(log₁₀ x)^5 = 56·x²·(log₁₀ x)^5</p><p><strong>Step 3:</strong> Set equal to 5600: 56·x²·(log₁₀ x)^5 = 5600</p><p><strong>Step 4:</strong> Divide by 56: x²·(log₁₀ x)^5 = 100</p><p><strong>Step 5:</strong> Test x = 10: (10)²·(log₁₀ 10)^5 = 100·(1)^5 = 100 ✓</p><p>∴ Answer: C (x = 10)</p>
Correct Answer: C