## Expanding (x/3 + 2/y)^4 with the Binomial Theorem

The binomial theorem provides a powerful tool for expanding expressions of the form (a + b)^n. Let's apply it to the expression (x/3 + 2/y)^4.

### The Binomial Theorem

The binomial theorem states:

**(a + b)^n = ∑_(k=0)^n ( n_C_k) a^(n-k) b^k**

Where:

**n**is a non-negative integer.**k**is an integer ranging from 0 to n.**(**is the binomial coefficient, calculated as n! / (k! * (n-k)!). This represents the number of ways to choose k objects from a set of n objects.*n_C_k*)

### Applying the Binomial Theorem to (x/3 + 2/y)^4

Let's identify our 'a' and 'b' terms:

**a = x/3****b = 2/y**

Now, we can apply the binomial theorem for n = 4:

**(x/3 + 2/y)^4 = ∑_(k=0)^4 ( 4_C_k) (x/3)^(4-k) (2/y)^k**

Let's expand the summation:

**(x/3 + 2/y)^4 = ( 4_C_0) (x/3)^4 (2/y)^0 + (4_C_1) (x/3)^3 (2/y)^1 + (4_C_2) (x/3)^2 (2/y)^2 + (4_C_3) (x/3)^1 (2/y)^3 + (4_C_4) (x/3)^0 (2/y)^4**

### Calculating the Binomial Coefficients and Simplifying

Now, let's calculate the binomial coefficients and simplify each term:

**(**: (x/3)^4 (2/y)^0 =*4_C_0*) = 1**x^4 / 81****(**: (x/3)^3 (2/y)^1 =*4_C_1*) = 4**8x^3 / 27y****(**: (x/3)^2 (2/y)^2 =*4_C_2*) = 6**24x^2 / 9y^2****(**: (x/3)^1 (2/y)^3 =*4_C_3*) = 4**32x / 3y^3****(**: (x/3)^0 (2/y)^4 =*4_C_4*) = 1**16 / y^4**

### The Final Expansion

Putting it all together, the complete expansion of (x/3 + 2/y)^4 is:

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

This is the expanded form of the given expression using the binomial theorem.