Combinatorics, EXT1 EQ-Bank 11 Find the term independent of \(x\) in the expansion of \(\left(2 x^3+\dfrac{1}{x^4}\right)^7\). (2 marks) --- 7 WORK AREA LINES (style=lined) --- Show Answers Only \(560\) Show Worked Solution \(T_k=\ \text {General term of} \ \ \left(2 x^3+\dfrac{1}{x^4}\right)^7\) \(T_k\) \(=\displaystyle \binom{7}{k}\left(2 x^3\right)^{7-k} \cdot\left(x^{-4}\right)^k\) \(=\displaystyle\binom{7}{k} \cdot 2^{7-k} \cdot x^{3(7-k)} \cdot x^{-4 k}\) \(=\displaystyle\binom{7}{k} \cdot 2^{7-k} \cdot x^{21-7 k}\) \(\text{Independent term occurs when:}\) \(x^{21-7 k}=x^0 \ \Rightarrow \ k=3\) \(\therefore \text{Independent term}=\displaystyle \binom{7}{3} \cdot 2^4=560\)