Question

Math

Posted about 1 month ago

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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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Answer from Sia

Verified answer

Posted about 1 month ago

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.

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