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

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

Question

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

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

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Binomial Theorem", "Binomial Expansion" ], "new_concepts": [], "current_concepts": [ "Binomial Theorem", "Binomial Expansion" ] } ## 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 $$ \begin{aligned} T_8 &= \binom{12}{7} (2x)^{12-7} (4y)^7\ &= 792 \cdot (2x)^5 \cdot (4y)^7\ &= 792 \cdot 32x^5 \cdot 16384y^7\ &= 792 \cdot 32 \cdot 16384 \cdot x^5 y^7\ &= 415,236,096 x^5 y^7 \end{aligned} $$ $415,236,096 x^5 y^7$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Binomial Theorem" ] }

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Question

Explanation:

Identify the binomial formula parameters

Determine the index for the eighth term

Calculate the eighth term

Answer: