%5E2-(4x-6y)%5E3-8x%5E3-27y%5E3.jpeg "(2x^2-3xy)^2 (4x-6y)^3 8x^3-27y^3")
Expansion of Algebraic Expressions
In this article, we will explore the expansion of algebraic expressions, specifically focusing on the expansion of (2x^2-3xy)^2 and (4x-6y)^3, and relate it to the expansion of 8x^3-27y^3.
Expansion of (2x^2-3xy)^2
To expand the expression (2x^2-3xy)^2, we need to follow the rule of squaring a binomial. The formula for squaring a binomial is:
(a + b)^2 = a^2 + 2ab + b^2
In this case, a = 2x^2 and b = -3xy. Substituting these values into the formula, we get:
(2x^2 - 3xy)^2 = (2x^2)^2 + 2(2x^2)(-3xy) + (-3xy)^2
Expanding the expression, we get:
(2x^2 - 3xy)^2 = 4x^4 - 12x^3y + 9x^2y^2
Expansion of (4x-6y)^3
To expand the expression (4x-6y)^3, we need to follow the rule of cubing a binomial. The formula for cubing a binomial is:
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
In this case, a = 4x and b = 6y. Substituting these values into the formula, we get:
(4x - 6y)^3 = (4x)^3 - 3(4x)^2(6y) + 3(4x)(6y)^2 - (6y)^3
Expanding the expression, we get:
(4x - 6y)^3 = 64x^3 - 288x^2y + 432xy^2 - 216y^3
Relationship with 8x^3-27y^3
Now, let's examine the expression 8x^3-27y^3. We can rewrite this expression as:
8x^3 - 27y^3 = (2x)^3 - (3y)^3
Using the difference of cubes formula, we can expand this expression as:
8x^3 - 27y^3 = (2x - 3y)((2x)^2 + (2x)(3y) + (3y)^2)
Simplifying the expression, we get:
8x^3 - 27y^3 = (2x - 3y)(4x^2 + 6xy + 9y^2)
Comparing this expression with the expanded forms of (2x^2 - 3xy)^2 and (4x - 6y)^3, we can see that they are related to each other. In fact, we can rewrite the expression 8x^3-27y^3 as a product of the expanded forms of (2x^2 - 3xy)^2 and (4x - 6y)^3.
This relationship highlights the importance of understanding algebraic expansions and their relationships with other expressions.
