(2x+3y)^6 Binomial Expansion

3 min read Jul 03, 2024

Binomial Expansion of (2x+3y)^6

In algebra, binomial expansion is a method of expanding powers of a binomial expression, which is an expression consisting of two terms. In this article, we will explore the binomial expansion of (2x+3y)^6.

The Binomial Theorem

The binomial theorem is a mathematical formula that describes the expansion of powers of a binomial expression. It states that:

(a+b)^n = a^n + na^(n-1)b + n(n-1)a^(n-2)b^2 + … + nab^(n-1) + b^n

where a and b are the two terms of the binomial expression, and n is a positive integer.

Expanding (2x+3y)^6

To expand (2x+3y)^6, we can use the binomial theorem with a = 2x, b = 3y, and n = 6. This gives us:

(2x+3y)^6 = (2x)^6 + 6(2x)^5(3y) + 15(2x)^4(3y)^2 + 20(2x)^3(3y)^3 + 15(2x)^2(3y)^4 + 6(2x)(3y)^5 + (3y)^6

Simplifying the Expansion

Now, let's simplify each term of the expansion:

(2x)^6 = 64x^6
6(2x)^5(3y) = 6(32x^5)(3y) = 576x^5y
15(2x)^4(3y)^2 = 15(16x^4)(9y^2) = 2160x^4y^2
20(2x)^3(3y)^3 = 20(8x^3)(27y^3) = 4320x^3y^3
15(2x)^2(3y)^4 = 15(4x^2)(81y^4) = 4860x^2y^4
6(2x)(3y)^5 = 6(2x)(243y^5) = 2916xy^5
(3y)^6 = 729y^6

Final Expansion

Therefore, the binomial expansion of (2x+3y)^6 is:

(2x+3y)^6 = 64x^6 + 576x^5y + 2160x^4y^2 + 4320x^3y^3 + 4860x^2y^4 + 2916xy^5 + 729y^6

This expansion can be useful in various mathematical and scientific applications, such as calculus, algebra, and physics.