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

Use Pascal's Triangle to expand $\left(x^{2}-z^{2}\right)^{4}$ . 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(x^{2}-z^{2}\right)^{4}

. Express your answer in simplest form.

\newline

Answer:

Identify Exponent Row: Identify the row of Pascal's Triangle that corresponds to the exponent $4$ . The $5$ th row (since we start counting from the $0$ th row for the exponent $0$ ) of Pascal's Triangle is $1, 4, 6, 4, 1$ . These numbers will be the coefficients in our expanded expression.
Write Binomial Theorem: Write out the terms of the expansion using the binomial theorem. $\newline$ The binomial theorem tells us that $(a - b)^n = \Sigma \binom{n}{k} \cdot a^{n-k} \cdot b^k$ , where $\Sigma$ denotes the sum over $k$ from $0$ to $n$ . For $(x^2 - z^2)^4$ , $a = x^2$ , $b = z^2$ , and $n = 4$ . We will use the coefficients from Pascal's Triangle to expand the expression.
Apply Coefficients: Apply the coefficients to each term of the expansion. $\newline$ The expanded form will be: $\newline$ $1\cdot(x^2)^4\cdot(z^2)^0 + 4\cdot(x^2)^3\cdot(z^2)^1 + 6\cdot(x^2)^2\cdot(z^2)^2 + 4\cdot(x^2)^1\cdot(z^2)^3 + 1\cdot(x^2)^0\cdot(z^2)^4$
Simplify Terms: Simplify each term of the expansion. $\newline$ Now we simplify the powers of $x^2$ and $z^2$ in each term: $\newline$ $1\cdot x^8\cdot z^0 + 4\cdot x^6\cdot z^2 + 6\cdot x^4\cdot z^4 + 4\cdot x^2\cdot z^6 + 1\cdot z^8$ $\newline$ Since $z^0 = 1$ , we can simplify further: $\newline$ $x^8 + 4\cdot x^6\cdot z^2 + 6\cdot x^4\cdot z^4 + 4\cdot x^2\cdot z^6 + z^8$
Final Simplified Expression: Write the final expanded expression in simplest form. $\newline$ The final expanded expression in simplest form is: $\newline$ $x^8 + 4x^6z^2 + 6x^4z^4 + 4x^2z^6 + z^8$

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