(4xy%5E5z%5E3)(-3x%5E5y%5E10z%5E18).jpeg "(2x^2y^2)(4xy^5z^3)(-3x^5y^10z^18)")
Simplifying Polynomial Expressions: A Step-by-Step Guide
This article will guide you through the process of simplifying the polynomial expression: (2x²y²)(4xy⁵z³)(-3x⁵y¹⁰z¹⁸).
Understanding the Basics
Before we dive into the simplification, let's clarify some fundamental concepts:
- Monomials: A monomial is a single term that can be a constant, a variable, or a product of constants and variables. For example, 2x², 4xy⁵z³, and -3x⁵y¹⁰z¹⁸ are all monomials.
- Coefficients: The numerical factor in a monomial is called the coefficient. In our example, the coefficients are 2, 4, and -3.
- Exponents: The exponents indicate how many times a variable is multiplied by itself. For instance, x² means x multiplied by itself twice (x * x).
Simplifying the Expression
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Rearrange the terms: We can rearrange the terms in the expression without changing the result. Let's group together the coefficients and the variables with the same base:
(2 * 4 * -3) (x² * x * x⁵) (y² * y⁵ * y¹⁰) (z³ * z¹⁸)
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Multiply the coefficients: Multiply the numerical coefficients:
-24 (x² * x * x⁵) (y² * y⁵ * y¹⁰) (z³ * z¹⁸)
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Apply the product of powers rule: When multiplying exponents with the same base, add the powers:
-24x⁸y¹⁷z²¹
Final Result
The simplified form of the expression (2x²y²)(4xy⁵z³)(-3x⁵y¹⁰z¹⁸) is -24x⁸y¹⁷z²¹.
Key Points
- Always remember the order of operations (PEMDAS/BODMAS).
- Be mindful of signs (positive and negative).
- Use the product of powers rule correctly when multiplying exponents with the same base.
- The final answer should be written in its simplest form with no redundant operations.