use pascal's triangle to expand the expression
(2x-3y)^3
the coefficients are 1,3,3,1 so you have
1(2x)^3(-3y)^0 + 3(2x)^2(-3y)^1 + 3(2x)^1(-3y)^2 + 1(2x)^0(-3y)^3
= 8x^3 - 36x^2y + 54xy^2 - 27y^3
thank you
To expand the expression (2x-3y)^3 using Pascal's triangle, we can use the binomial theorem.
The binomial theorem states that for any positive integer n, the expansion of (a+b)^n can be found by adding the terms obtained by raising "a" to the power of n, n-1, n-2, ..., 2, 1, and raising "b" to the power of 0, 1, 2, ..., n-1, n, respectively, and multiplying each term by the corresponding binomial coefficient.
In this case, we have (2x-3y)^3, so a = 2x and b = -3y. We need to expand it up to the power of 3.
The binomial coefficients can be found in Pascal's triangle, where each number is the sum of the two numbers directly above it.
Let's start expanding using the binomial theorem:
For the power of 3, we have 4 terms: (2x)^3, (2x)^2 * (-3y), (2x) * (-3y)^2, and (-3y)^3.
The binomial coefficients we need to multiply with each term are found in the third row of Pascal's triangle: 1, 3, 3, 1.
(2x-3y)^3 = 1 * (2x)^3 + 3 * (2x)^2 * (-3y) + 3 * (2x) * (-3y)^2 + 1 * (-3y)^3
Next, we simplify each term:
(2x)^3 = (2x)(2x)(2x) = 8x^3
(2x)^2 = (2x)(2x) = 4x^2
(-3y)^2 = (-3y)(-3y) = 9y^2
(-3y)^3 = (-3y)(-3y)(-3y) = -27y^3
Substituting these values back into our expression:
(2x-3y)^3 = 1 * 8x^3 + 3 * 4x^2 * (-3y) + 3 * (2x) * 9y^2 + 1 * (-27y^3)
Simplifying further:
(2x-3y)^3 = 8x^3 - 36x^2y + 54xy^2 - 27y^3
So, the expanded expression of (2x-3y)^3 using Pascal's triangle is 8x^3 - 36x^2y + 54xy^2 - 27y^3.