(-3x%5E2y%5E7).jpeg "(2x^4y^5)(-3x^2y^7)")
Multiplying Monomials: A Step-by-Step Guide
This article will guide you through the process of multiplying the monomials $(2x^4y^5)(-3x^2y^7)$.
Understanding Monomials
Monomials are algebraic expressions consisting of a single term. They are formed by multiplying constants and variables raised to non-negative integer powers. For example, $2x^4y^5$ and $-3x^2y^7$ are both monomials.
The Multiplication Process
To multiply monomials, we follow these steps:
- Multiply the coefficients: In our case, this is $2 \times (-3) = -6$.
- Multiply the variables with the same base by adding their exponents:
- For x, we have $x^4 \times x^2 = x^{4+2} = x^6$.
- For y, we have $y^5 \times y^7 = y^{5+7} = y^{12}$.
Final Result
Combining the results from steps 1 and 2, we get the final product:
$(2x^4y^5)(-3x^2y^7) = \boxed{-6x^6y^{12}}$
Key Points to Remember:
- Commutative Property of Multiplication: The order of multiplication doesn't affect the result.
- Associative Property of Multiplication: Grouping factors differently doesn't change the result.
- Exponent Rules: When multiplying variables with the same base, we add their exponents.
By understanding these concepts, you can confidently multiply monomials and simplify complex algebraic expressions.