Independent Term D64514

1. **State the problem:** Find the term independent of $x$ in the expansion of $$\left(x-\frac{2}{x}\right)^6$$. 2. **Formula used:** Use the Binomial Theorem: $$\left(a+b\right)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$$ Here, $a = x$ and $b = -\frac{2}{x}$, and $n=6$. 3. **General term:** The $k$-th term (starting from $k=0$) is $$T_{k+1} = \binom{6}{k} x^{6-k} \left(-\frac{2}{x}\right)^k = \binom{6}{k} (-2)^k x^{6-k-k} = \binom{6}{k} (-2)^k x^{6-2k}$$ 4. **Find term independent of $x$:** The power of $x$ must be zero: $$6 - 2k = 0 \implies 2k = 6 \implies k = 3$$ 5. **Calculate the term for $k=3$:** $$T_4 = \binom{6}{3} (-2)^3 x^{6-6} = 20 \times (-8) \times x^0 = -160$$ 6. **Answer:** The term independent of $x$ is **$-160$**.