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Find the 2nd term in the expansion of
(x-8)^(8) in simplest form.
Answer:

Find the $2$ nd term in the expansion of $(x-8)^{8}$ in simplest form. $\newline$ Answer:

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Evaluate rational expressions II

Full solution

Q. Find the

2

nd term in the expansion of

(x-8)^{8}

in simplest form.

\newline

Answer:

Use Binomial Theorem: To find the $2$ nd term in the expansion of $(x-8)^{8}$ , we will use the binomial theorem. The general form of the $k$ -th term in the expansion of $(a+b)^n$ is given by $T(k) = C(n, k-1) \cdot a^{(n-k+1)} \cdot b^{(k-1)}$ , where $C(n, k)$ is the binomial coefficient ＂ $n$ choose $k$ ＂. For the $2$ nd term, $k = 2$ .
Calculate Binomial Coefficient: Calculate the binomial coefficient for the $2$ nd term, which is $C(8, 2-1) = C(8, 1)$ . The binomial coefficient $C(n, k)$ is calculated as $\frac{n!}{k! \cdot (n-k)!}$ , where $\text{“!“}$ denotes factorial.
Compute $C(8, 1)$ : Compute $C(8, 1)$ using the formula for binomial coefficients. $C(8, 1) = \frac{8!}{1! \cdot (8-1)!} = \frac{8}{1} = 8$ .
Calculate Powers of a and b: Now, we need to calculate the rest of the $2$ nd term using the powers of $a$ and $b$ . In our case, $a = x$ and $b = -8$ . For the $2$ nd term, $a^{(n-k+1)} = x^{(8-2+1)} = x^{7}$ and $b^{(k-1)} = (-8)^{(2-1)} = -8$ .
Combine to Get $2$ nd Term: Combine the binomial coefficient with the powers of $a$ and $b$ to get the $2$ nd term. The $2$ nd term is $8 \times x^{7} \times (-8) = -64 \times x^{7}$ .

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