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Chapter 7: Problem 2

Multiply. (Assume all denominators are nonzero.) $$ -5 x 3 y \cdot y 225 x $$

Short Answer

Expert verified
\(-3375x^2y^2\)

Step by step solution

01

Rearrange the Expression

Rewrite the expression \(-5x3y \cdot 225xy\) to have all coefficients, variables, and their powers grouped together. Combine similar terms: \((-5 \cdot 225) \cdot (x \cdot x) \cdot (3y \cdot y)\).
02

Multiply the Coefficients

Multiply the numerical coefficients: \(-5 \times 3 \times 225 = -3375\).
03

Apply the Laws of Exponents

For the variables, combine the powers: \(x \cdot x = x^2\) and \(3y \cdot y = 3y^2\).
04

Combine the Results

Multiply the combined coefficients and the simplified variable expressions to get \(-3375x^2y^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Exponents
Exponents are a mathematical way of expressing repeated multiplication. When you see an expression like \(x^2\), it means \(x\) is multiplied by itself, that is \(x \times x\). Understanding how to work with exponents is crucial when multiplying terms that contain the same base.
  • Same Base Rule: When multiplying terms that have the same base, for example \(x^a \times x^b\), you can simply add the exponents. This means \(x^a \times x^b = x^{a+b}\).
  • Examples: In the expression \(x \cdot x\), the base \(x\) is common in both terms, so we add the exponents: \(x^1 \cdot x^1 = x^{1+1} = x^2\).
It's important to write all variables in a clear and organized way. This helps you track which exponents are added together.
Rearranging Expressions
Rearranging expressions is a helpful strategy to simplify and organize terms before performing operations. By grouping similar terms and organizing by coefficients and variables, you can make the multiplication process much easier.
  • Purpose: The main goal is to put order in expressions, making it simpler to perform operations like multiplication.
  • Step-by-Step: Start by identifying all the like terms and coefficients. Then, group these terms together. For example, in the expression \(-5x3y \cdot 225xy\), you rearrange it as \((-5 \cdot 225) \cdot (x \cdot x) \cdot (3y \cdot y)\).

Reorganizing helps reduce mistakes and ensures you're combining and manipulating terms correctly.
Multiplying Coefficients
Multiplying coefficients involves handling the numerical part of an expression separately from the variables. This is a straightforward yet essential step in simplifying products, especially in polynomial multiplication.
  • Definition: Coefficients are the constant numerical values that multiply variables within a term.
  • How to Multiply: Simply multiply these coefficients as you would with any numbers. For instance, with \(-5 \times 3 \times 225\), you multiply each step by step to get \(-3375\).
By separating and focusing first on the coefficients and then on exponents and variables, you can systematically work through complex polynomial multiplications.

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