Final answer:
The fourth term of the expansion (3c+4d)⁷ is calculated using the binomial theorem. The term is found using the formula T(n, k) which gives 22680c⁴d³ as the fourth term.
Step-by-step explanation:
The fourth term of the expansion (3c+4d)⁷ using the binomial theorem is found by using the general term formula for a binomial expansion: T(n, k) = C(n, k) • a²(n-k) • bⁿ where n is the power of the binomial, k is the term number minus 1, a and b are the terms of the binomial, and C(n, k) is the binomial coefficient "n choose k". For the fourth term (k=3), since the first term is considered k=0, the formula becomes C(7, 3) • (3c)⁴ • (4d)³.
Calculating this we get:
- C(7, 3) = 7! / [3! • (7-3)!] = 7 • 6 • 5 / (3 • 2 • 1) = 35
- (3c)⁴ = 81•c⁴
- (4d)³ = 64•d³
Multiplying these together:
35 • 81c⁴ • 64d³ = 22680c⁴d³
So, the fourth term of the expansion (3c+4d)⁷ is 22680c⁴d³.