3. Find the coefficient of $x^{-4}$ in the expansion of $\left(\frac{x^{2}}{2}-\frac{4}{x^{3}}\right)^{8}$.

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∻Solution by Steps∻ ‖ step 1 ⋮ We start with the expression $ \left( \frac{x^{2}}{2} - \frac{4}{x^{3}} \right)^{8} $ and need to find the coefficient of $ x^{-4} $. ‖ ‖ step 2 ⋮ Using the binomial theorem, we can expand this expression as $ \sum_{k=0}^{8} \binom{8}{k} \left( \frac{x^{2}}{2} \right)^{8-k} \left( -\frac{4}{x^{3}} \right)^{k} $. ‖ ‖ step 3 ⋮ This simplifies to $ \sum_{k=0}^{8} \binom{8}{k} \frac{x^{2(8-k)}}{2^{8-k}} \cdot \frac{(-4)^{k}}{x^{3k}} $. ‖ ‖ step 4 ⋮ We need to find $ k $ such that the exponent of $ x $ is $ -4 $: $ 2(8-k) - 3k = -4 $. Solving this gives $ 16 - 2k - 3k = -4 $ or $ 16 - 5k = -4 $, leading to $ 5k = 20 $ and $ k = 4 $. ‖ ‖ step 5 ⋮ Now, substituting $ k = 4 $ back into the binomial expansion, we find the coefficient: $ \binom{8}{4} \frac{(-4)^{4}}{2^{4}} $. ‖ ‖ step 6 ⋮ Calculating $ \binom{8}{4} = 70 $, $ (-4)^{4} = 256 $, and $ 2^{4} = 16 $, we have $ 70 \cdot \frac{256}{16} = 70 \cdot 16 = 1120 $. ‖ ∻Answer∻ ⚹ The coefficient of $ x^{-4} $ in the expansion is $ 1120 $. ⚹ ∻Key Concept∻ ⚹ The binomial theorem allows us to expand expressions of the form $ (a + b)^n $ and find specific coefficients in the expansion. ⚹ ∻Explanation∻ ⚹ By applying the binomial theorem and solving for the appropriate $ k $, we can determine the coefficient of $ x^{-4} $ in the given expression. ⚹◊What is the general formula for finding the coefficient of $ x^n $ in the expansion of $ (a + b)^k $ using the Binomial Theorem?⍭ Generate me a similar question◊

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