(X+3y)^4
To expand the expression (x + 3y)^4, we can use the binomial theorem.
The binomial theorem states that (a + b)^n can be expanded as the sum of terms where each term is of the form (n choose k) * a^(n-k) * b^k, where (n choose k) is the binomial coefficient and can be calculated as n! / (k! * (n-k)!).
In this case, a = x, b = 3y, and n = 4.
So, applying the binomial theorem, we have:
(x + 3y)^4 = (4 choose 0) * x^4 * (3y)^0 + (4 choose 1) * x^3 * (3y)^1 + (4 choose 2) * x^2 * (3y)^2 + (4 choose 3) * x^1 * (3y)^3 + (4 choose 4) * x^0 * (3y)^4
Simplifying the terms, we get:
(1) * x^4 * 1 + (4) * x^3 * 3y + (6) * x^2 * (3y)^2 + (4) * x^1 * (3y)^3 + (1) * x^0 * (3y)^4
This can be further simplified as:
x^4 + 12x^3y + 54x^2y^2 + 108xy^3 + 81y^4
Therefore, the expression (x + 3y)^4 expands to x^4 + 12x^3y + 54x^2y^2 + 108xy^3 + 81y^4.
(2x-4)^5
To expand the expression (2x-4)^5, we can use the binomial theorem.
The binomial theorem states that (a + b)^n can be expanded as the sum of terms where each term is of the form (n choose k) * a^(n-k) * b^k, where (n choose k) is the binomial coefficient and can be calculated as n! / (k! * (n-k)!).
In this case, a = 2x, b = -4, and n = 5.
So, applying the binomial theorem, we have:
(2x - 4)^5 = (5 choose 0) * (2x)^5 * (-4)^0 + (5 choose 1) * (2x)^4 * (-4)^1 + (5 choose 2) * (2x)^3 * (-4)^2 + (5 choose 3) * (2x)^2 * (-4)^3 + (5 choose 4) * (2x)^1 * (-4)^4 + (5 choose 5) * (2x)^0 * (-4)^5
Simplifying the terms, we get:
(1) * (2x)^5 * 1 + (5) * (2x)^4 * (-4) + (10) * (2x)^3 * 16 + (10) * (2x)^2 * (-64) + (5) * (2x)^1 * 256 + (1) * (2x)^0 * (-1024)
This can be further simplified as:
32x^5 - 320x^4 + 1280x^3 - 2560x^2 + 2560x - 1024
Therefore, the expression (2x-4)^5 expands to 32x^5 - 320x^4 + 1280x^3 - 2560x^2 + 2560x - 1024.
To expand the expression (x + 3y)^4, you can use the binomial theorem or the Pascal's triangle method. Here's how you can do it using the binomial theorem:
Step 1: Identify the values of n, a, b, and c in the binomial theorem.
In this case, n = 4, a = x, and b = 3y.
Step 2: Write out the binomial coefficients.
The binomial coefficients for n = 4 are 1, 4, 6, 4, and 1.
Step 3: Expand and simplify each term using the formula of the binomial theorem:
(x + 3y)^4 = (1C0)(x^4)(3y)^0 + (1C1)(x^3)(3y)^1 + (1C2)(x^2)(3y)^2 + (1C3)(x^1)(3y)^3 + (1C4)(x^0)(3y)^4
Step 4: Simplify each term:
(1)(x^4)(1) + (4)(x^3)(3y) + (6)(x^2)(9y^2) + (4)(x)(27y^3) + (1)(1)(81y^4)
Step 5: Simplify further:
x^4 + 12x^3y + 54x^2y^2 + 108xy^3 + 81y^4
Therefore, (x + 3y)^4 expands to x^4 + 12x^3y + 54x^2y^2 + 108xy^3 + 81y^4.