(x/3+2/y)^4 In Binomial Theorem

3 min read Jun 17, 2024

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.
(n_C_k) 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.

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:

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