Expand (x/3 + 2/y)4

Sure, let's expand the expression (x/3 + 2/y)^4 step by step:

Step 1: Write down the binomial expansion formula which is (a + b)^n = Σ (n choose k) * a^(n-k) * b^k from k=0 to n. Here, n is the power of the binomial, in this case 4.

Step 2: Identify a and b in the expression. Here, a = x/3 and b = 2/y.

Step 3: Substitute a, b, and n into the binomial expansion formula:

(x/3 + 2/y)^4 = Σ (4 choose k) * (x/3)^(4-k) * (2/y)^k from k=0 to 4.

Step 4: Expand the summation:

= (4 choose 0) * (x/3)^4 * (2/y)^0 + (4 choose 1) * (x/3)^3 * (2/y)^1 + (4 choose 2) * (x/3)^2 * (2/y)^2 + (4 choose 3) * (x/3)^1 * (2/y)^3 + (4 choose 4) * (x/3)^0 * (2/y)^4.

Step 5: Simplify each term:

= 1 * x^4 / 81 * 1 + 4 * x^3 / 27 * 2/y + 6 * x^2 / 9 * 4/y^2 + 4 * x / 3 * 8/y^3 + 1 * 16/y^4.

Step 6: Simplify further:

= x^4 / 81 + 8x^3 / 27y + 24x^2 / 9y^2 + 32x / 3y^3 + 16/y^4.

So, the expansion of (x/3 + 2/y)^4 is x^4 / 81 + 8x^3 / 27y + 24x^2 / 9y^2 + 32x / 3y^3 + 16/y^4.