To determine which row of Pascal's triangle will be used for expanding the given binomial expression [tex]\(\left(2x^3 + 3y^2\right)^7\)[/tex], follow these steps:
1. Understand the Binomial Theorem: The Binomial Theorem states that [tex]\((a + b)^n\)[/tex] can be expanded as:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
Here, [tex]\(\binom{n}{k}\)[/tex] are the binomial coefficients.
2. Identify [tex]\( n \)[/tex] in the Expression: In the given binomial expression [tex]\(\left(2x^3 + 3y^2\right)^7\)[/tex], we see that the exponent [tex]\( n \)[/tex] is 7. This means we need the 7th row of Pascal’s triangle.
3. Pascal's Triangle and Row Identification: Pascal's triangle is constructed with each row representing the coefficients of the expanded form of [tex]\((a + b)^n\)[/tex]. The [tex]\( n \)[/tex]-th row (starting with [tex]\( n = 0 \)[/tex] for the top row) contains the binomial coefficients [tex]\(\binom{n}{k}\)[/tex] for [tex]\( k = 0 \)[/tex] to [tex]\( k = n \)[/tex].
4. Retrieve the Correct Row: Specifically, the 7th row of Pascal's triangle gives us the coefficients for [tex]\((a + b)^7\)[/tex]. The 7th row of Pascal’s triangle is:
[tex]\[
[1, 7, 21, 35, 35, 21, 7, 1]
\][/tex]
Hence, for expanding the binomial expression [tex]\(\left(2 x^3 + 3 y^2\right)^7\)[/tex], the row of Pascal's triangle that will be used is the 7th row comprising the coefficients [tex]\([1, 7, 21, 35, 35, 21, 7, 1]\)[/tex].
