Detailed Overview of Multiplying a Monomial by a Monomial
In this section, we focus on the foundational operation of multiplying monomials, which are algebraic expressions that contain only one term.
Definition
A monomial is defined as an expression that includes only one term, upon which we can perform multiplication in a systematic manner. For example, monomials may appear as simple numbers (like 4 or -3) or variable expressions (like 3xy or -15abc).
Key Points:
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Multiplying Two Monomials: The product of two monomials results in another monomial. The general rule involves multiplying the coefficients (numerical parts) while applying the rules of exponents for the variable parts.
- Examples:
- If we multiply
x with 3y, we write:
x × 3y = 3xy.
- Multiplying
5x with 4x², we find:
5x × 4x² = (5 × 4) × (x × x²) = 20x³.
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Example with Negative Coefficients: For instance,
5x × (–3y) gives us –15xy.
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Multiplying Three Monomials: The multiplication rules extend to three or more monomials. Combine coefficients first, and then variables:
2x × 5y × 7z = (2 × 5 × 7)(x × y × z) = 70xyz.
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The associative property allows various groupings during multiplication, giving us consistency regardless of how we organize the expressions for calculations.
Conclusion
Understanding how to multiply monomials sets the groundwork for more advanced operations in algebra, such as working with polynomials, where the same foundational principles apply.