Final answer:
The coefficient of x³y⁴ in the expression (-3x⁴y)⁷ is 0 because the expansion of this expression does not contain a term with the power of x being 3.
Step-by-step explanation:
To find the coefficient of x³y⁴ in the expression (-3x⁴y)⁷, we have to expand the expression using the binomial theorem, which tells us how to expand expressions of the form (a+b)⁷. However, since there is no direct + or - between terms in (-3x⁴y)⁷, we do not actually use the binomial theorem but raise each term to the seventh power. This involves raising both -3 and x⁴y to the seventh power, which means we will multiply the exponent of x and y by 7.
The power of x in x⁴ raised to the 7th power would become x^(4*7) = x²⁸. Since we are looking for the term where the power of x is 3, which is not possible in this expansion, we can conclude that there is no term with x³y⁴ in (-3x⁴y)⁷, and therefore, the coefficient of x³y⁴ is 0.