Question

determine the eighth term of the following expression without expanding the binomial.

\\(2x + 4y)^{12}\\

Explanation:

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<reasoning>

Identify the binomial formula parameters

\[

$$\begin{aligned} &(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\\ &a = 2x,\quad b = 4y,\quad n = 12 \end{aligned}$$

\]

Determine the index for the eighth term

\[

$$\begin{aligned} &T_{r+1} = \binom{n}{r} a^{n-r} b^r\\ &r + 1 = 8 \implies r = 7 \end{aligned}$$

\]

Calculate the eighth term

\[

$$\begin{aligned} T_8 &= \binom{12}{7} (2x)^{12-7} (4y)^7\\ &= 792 \cdot (2x)^5 \cdot (4y)^7\\ &= 792 \cdot 32x^5 \cdot 16384y^7\\ &= 792 \cdot 32 \cdot 16384 \cdot x^5 y^7\\ &= 415,236,096 x^5 y^7 \end{aligned}$$

\]
</reasoning>

<answer>
\(415,236,096 x^5 y^7\)
</answer>

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Answer:

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<reasoning>

Identify the binomial formula parameters

\[

$$\begin{aligned} &(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\\ &a = 2x,\quad b = 4y,\quad n = 12 \end{aligned}$$

\]

Determine the index for the eighth term

\[

$$\begin{aligned} &T_{r+1} = \binom{n}{r} a^{n-r} b^r\\ &r + 1 = 8 \implies r = 7 \end{aligned}$$

\]

Calculate the eighth term

\[

$$\begin{aligned} T_8 &= \binom{12}{7} (2x)^{12-7} (4y)^7\\ &= 792 \cdot (2x)^5 \cdot (4y)^7\\ &= 792 \cdot 32x^5 \cdot 16384y^7\\ &= 792 \cdot 32 \cdot 16384 \cdot x^5 y^7\\ &= 415,236,096 x^5 y^7 \end{aligned}$$

\]
</reasoning>

<answer>
\(415,236,096 x^5 y^7\)
</answer>

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