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Find the 2 nd term in the expansion of
(2x-9y)^(7) in simplest form.
Answer:

Find the $2$ nd term in the expansion of $(2 x-9 y)^{7}$ in simplest form. $\newline$ Answer:

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Evaluate rational exponents

Full solution

Q. Find the

2

nd term in the expansion of

(2 x-9 y)^{7}

in simplest form.

\newline

Answer:

Identify Binomial Expansion Form: Identify the general form of the binomial expansion. $\newline$ The binomial theorem states that $(a + b)^n$ expands to a sum of terms of the form $C(n, k) \cdot a^{(n-k)} \cdot b^k$ , where $C(n, k)$ is the binomial coefficient, equal to $\frac{n!}{k!(n-k)!}$ .
Determine Second Term: Determine the second term in the expansion. $\newline$ The second term corresponds to $k = 1$ , so we will calculate $C(7, 1) \cdot (2x)^{7-1} \cdot (-9y)^1$ .
Calculate Binomial Coefficient: Calculate the binomial coefficient $C(7, 1)$ . $C(7, 1) = \frac{7!}{(1!(7-1)!)} = \frac{7!}{(1! \cdot 6!)} = \frac{7}{1} = 7$ .
Calculate Powers: Calculate the powers of $(2x)$ and $(-9y)$ . $(2x)^{7-1} = (2x)^6$ and $(-9y)^1 = -9y$ .
Multiply Terms: Multiply the binomial coefficient by the powers of $(2x)$ and $(-9y)$ . The second term is $7 \times (2x)^6 \times (-9y) = 7 \times 64x^6 \times (-9y) = -4032x^6y$ .

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