Answer :
To find the coefficient of the [tex]\(x^9 y\)[/tex] term in the binomial expansion of [tex]\((2y + 4x^3)^4\)[/tex], we can follow these steps:
1. Identify the given terms and the required term:
- We are expanding [tex]\((2y + 4x^3)^4\)[/tex].
- We need the term that has [tex]\(x^9 y\)[/tex].
2. Understand how binomial expansion works:
- The binomial expansion of [tex]\((a + b)^n\)[/tex] is given by:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
- Here, [tex]\(a = 2y\)[/tex] and [tex]\(b = 4x^3\)[/tex].
3. Determine the powers needed to get [tex]\(x^9\)[/tex]:
- Since each [tex]\(b = 4x^3\)[/tex] contains [tex]\(x^3\)[/tex], to get [tex]\(x^9\)[/tex], we need [tex]\((x^3)^3\)[/tex].
- So, we choose [tex]\(k = 3\)[/tex].
4. Calculate the corresponding term's coefficient:
- The binomial coefficient is [tex]\(\binom{4}{3}\)[/tex].
- Calculate [tex]\(\binom{4}{3} = 4\)[/tex].
5. Compute the specific term:
- According to the binomial expansion formula, we find the term with [tex]\(k = 3\)[/tex] as:
[tex]\[
\text{Term} = \binom{4}{3} \cdot (2y)^{4-3} \cdot (4x^3)^3
\][/tex]
- Simplify this:
- [tex]\((2y)^1 = 2y\)[/tex]
- [tex]\((4x^3)^3 = 4^3 \cdot (x^3)^3 = 64x^9\)[/tex]
6. Combine the coefficients:
- The full term becomes:
[tex]\[
4 \cdot 2y \cdot 64x^9 = 512x^9y
\][/tex]
Thus, the coefficient of the [tex]\(x^9 y\)[/tex] term is [tex]\(512\)[/tex].
1. Identify the given terms and the required term:
- We are expanding [tex]\((2y + 4x^3)^4\)[/tex].
- We need the term that has [tex]\(x^9 y\)[/tex].
2. Understand how binomial expansion works:
- The binomial expansion of [tex]\((a + b)^n\)[/tex] is given by:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
- Here, [tex]\(a = 2y\)[/tex] and [tex]\(b = 4x^3\)[/tex].
3. Determine the powers needed to get [tex]\(x^9\)[/tex]:
- Since each [tex]\(b = 4x^3\)[/tex] contains [tex]\(x^3\)[/tex], to get [tex]\(x^9\)[/tex], we need [tex]\((x^3)^3\)[/tex].
- So, we choose [tex]\(k = 3\)[/tex].
4. Calculate the corresponding term's coefficient:
- The binomial coefficient is [tex]\(\binom{4}{3}\)[/tex].
- Calculate [tex]\(\binom{4}{3} = 4\)[/tex].
5. Compute the specific term:
- According to the binomial expansion formula, we find the term with [tex]\(k = 3\)[/tex] as:
[tex]\[
\text{Term} = \binom{4}{3} \cdot (2y)^{4-3} \cdot (4x^3)^3
\][/tex]
- Simplify this:
- [tex]\((2y)^1 = 2y\)[/tex]
- [tex]\((4x^3)^3 = 4^3 \cdot (x^3)^3 = 64x^9\)[/tex]
6. Combine the coefficients:
- The full term becomes:
[tex]\[
4 \cdot 2y \cdot 64x^9 = 512x^9y
\][/tex]
Thus, the coefficient of the [tex]\(x^9 y\)[/tex] term is [tex]\(512\)[/tex].
