%5E4.jpeg "(x-7y)^4")
Expanding (x-7y)^4
The expression (x-7y)^4 represents the product of (x-7y) multiplied by itself four times:
(x-7y)^4 = (x-7y) * (x-7y) * (x-7y) * (x-7y)
To expand this expression, we can use the binomial theorem or the distributive property.
Binomial Theorem Approach
The binomial theorem provides a formula for expanding expressions of the form (a+b)^n:
(a + b)^n = Σ (n choose k) * a^(n-k) * b^k
Where:
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!).
- Σ represents the sum from k = 0 to n.
Applying this to our expression:
(x - 7y)^4 = Σ (4 choose k) * x^(4-k) * (-7y)^k
Expanding the sum:
(x - 7y)^4 = (4 choose 0) * x^4 * (-7y)^0 + (4 choose 1) * x^3 * (-7y)^1 + (4 choose 2) * x^2 * (-7y)^2 + (4 choose 3) * x^1 * (-7y)^3 + (4 choose 4) * x^0 * (-7y)^4
Calculating the binomial coefficients and simplifying:
(x - 7y)^4 = x^4 + -28x^3y + 294x^2y^2 + -1372xy^3 + 2401y^4
Distributive Property Approach
We can also expand the expression by repeatedly applying the distributive property:
- (x-7y) * (x-7y) = x^2 - 14xy + 49y^2
- (x^2 - 14xy + 49y^2) * (x-7y) = x^3 - 21x^2y + 147xy^2 - 343y^3
- (x^3 - 21x^2y + 147xy^2 - 343y^3) * (x-7y) = x^4 - 28x^3y + 294x^2y^2 - 1372xy^3 + 2401y^4
Both methods lead to the same expanded form:
x^4 - 28x^3y + 294x^2y^2 - 1372xy^3 + 2401y^4