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Use Pascal's Triangle to expand
(3z^(2)+5)^(4). Express your answer in simplest form.
Answer:

Use Pascal's Triangle to expand $\left(3 z^{2}+5\right)^{4}$ . Express your answer in simplest form. $\newline$ Answer:

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Pascal's triangle and the Binomial Theorem

Full solution

Q. Use Pascal's Triangle to expand

\left(3 z^{2}+5\right)^{4}

. Express your answer in simplest form.

\newline

Answer:

Identify Row: Identify the row of Pascal's Triangle that corresponds to the exponent $4$ . The $5$ th row (since we start counting from $0$ ) of Pascal's Triangle is $1, 4, 6, 4, 1$ . These numbers will be the coefficients in our expanded expression.
Write Expansion Terms: Write out the terms of the expansion using the binomial theorem and the coefficients from Pascal's Triangle. $\newline$ The binomial theorem tells us that $(a + b)^n = \Sigma (n \choose k) \cdot a^{n-k} \cdot b^k$ , where $\Sigma$ denotes the sum over $k$ from $0$ to $n$ . $\newline$ For $(3z^{2} + 5)^4$ , we have: $\newline$ $1\cdot(3z^{2})^4\cdot(5)^0 + 4\cdot(3z^{2})^3\cdot(5)^1 + 6\cdot(3z^{2})^2\cdot(5)^2 + 4\cdot(3z^{2})^1\cdot(5)^3 + 1\cdot(3z^{2})^0\cdot(5)^4$
Calculate Each Term: Calculate each term of the expansion. $\newline$ $1\times(3z^{2})^{4}\times(5)^{0} = 1\times(81z^{8})\times(1) = 81z^{8}$ $\newline$ $4\times(3z^{2})^{3}\times(5)^{1} = 4\times(27z^{6})\times(5) = 540z^{6}$ $\newline$ $6\times(3z^{2})^{2}\times(5)^{2} = 6\times(9z^{4})\times(25) = 1350z^{4}$ $\newline$ $4\times(3z^{2})^{1}\times(5)^{3} = 4\times(3z^{2})\times(125) = 1500z^{2}$ $\newline$ $1\times(3z^{2})^{0}\times(5)^{4} = 1\times(1)\times(625) = 625$
Combine for Final Form: Combine all the terms to write the final expanded form. $\newline$ The expanded form of $(3z^{2} + 5)^4$ is: $\newline$ $81z^{8} + 540z^{6} + 1350z^{4} + 1500z^{2} + 625$

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