(9%2Fx%5E2%2B25%2Fy%5E2%2B15%2Fxy).jpeg "(3/x-5/y)(9/x^2+25/y^2+15/xy)")
Simplifying Algebraic Expressions: (3/x - 5/y)(9/x^2 + 25/y^2 + 15/xy)
This article will guide you through the process of simplifying the algebraic expression: (3/x - 5/y)(9/x^2 + 25/y^2 + 15/xy).
Understanding the Expression
The given expression is a product of two factors:
- (3/x - 5/y): This is a binomial with two terms, each representing a fraction with variables x and y in the denominator.
- (9/x^2 + 25/y^2 + 15/xy): This is a trinomial with three terms, each representing a fraction with variables x and y in the denominator.
Simplifying using the Distributive Property
To simplify this expression, we'll use the distributive property. This means we multiply each term in the first factor by each term in the second factor.
Let's break down the multiplication:
- (3/x) * (9/x^2): This gives us 27/x^3.
- (3/x) * (25/y^2): This gives us 75/xy^2.
- (3/x) * (15/xy): This gives us 45/x^2y.
- (-5/y) * (9/x^2): This gives us -45/x^2y.
- (-5/y) * (25/y^2): This gives us -125/y^3.
- (-5/y) * (15/xy): This gives us -75/xy^2.
Now, we combine the like terms:
27/x^3 + 75/xy^2 + 45/x^2y - 45/x^2y - 125/y^3 - 75/xy^2
Simplifying further, we get:
27/x^3 - 125/y^3
Conclusion
Therefore, the simplified form of the expression (3/x - 5/y)(9/x^2 + 25/y^2 + 15/xy) is 27/x^3 - 125/y^3. This process illustrates how distributive property and combining like terms can be used to simplify complex algebraic expressions.