(2x+3y)^5

3 min read Jun 16, 2024
(2x+3y)^5

Expanding (2x + 3y)^5 Using the Binomial Theorem

The binomial theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a positive integer. In this case, we want to expand (2x + 3y)^5.

Applying the Binomial Theorem

The binomial theorem states:

(a + b)^n = ∑ (n choose k) * a^(n-k) * b^k

Where:

  • n choose k is the binomial coefficient, calculated as n! / (k! * (n-k)!).
  • k ranges from 0 to n.

Let's apply this to our expression:

(2x + 3y)^5 = ∑ (5 choose k) * (2x)^(5-k) * (3y)^k

Expanding the Summation

We need to calculate the terms for each value of k from 0 to 5:

  • k = 0: (5 choose 0) * (2x)^5 * (3y)^0 = 1 * 32x^5 * 1 = 32x^5
  • k = 1: (5 choose 1) * (2x)^4 * (3y)^1 = 5 * 16x^4 * 3y = 240x^4y
  • k = 2: (5 choose 2) * (2x)^3 * (3y)^2 = 10 * 8x^3 * 9y^2 = 720x^3y^2
  • k = 3: (5 choose 3) * (2x)^2 * (3y)^3 = 10 * 4x^2 * 27y^3 = 1080x^2y^3
  • k = 4: (5 choose 4) * (2x)^1 * (3y)^4 = 5 * 2x * 81y^4 = 810xy^4
  • k = 5: (5 choose 5) * (2x)^0 * (3y)^5 = 1 * 1 * 243y^5 = 243y^5

Final Expanded Form

Combining all the terms, we get the complete expansion of (2x + 3y)^5:

(2x + 3y)^5 = 32x^5 + 240x^4y + 720x^3y^2 + 1080x^2y^3 + 810xy^4 + 243y^5

This demonstrates how the binomial theorem simplifies the expansion of complex expressions with exponents.

Related Post