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Chapter 2: Problem 4

Are the two expressions equivalent? \(2 x y\) and \((2 x)(2 y)\)

Short Answer

Expert verified
Answer: No, the expressions are not equivalent.

Step by step solution

01

Simplify the first expression

The first expression is already in its simplest form: \(2xy\).
02

Simplify the second expression

We can start by distributing \((2x)(2y)\): \((2x)(2y) = 2 * 2 * x * y = 4xy\)
03

Compare the simplified expressions

We can now compare both expressions to see if they are equivalent: The first expression is \(2xy\), while the second expression is \(4xy\). As these two expressions are not the same, they are not equivalent.

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations. In mathematics, variables like \(x\) or \(y\) represent unknown values or quantities that can change.
For example, the expression \(2xy\) is a common algebraic expression involving two variables \(x\) and \(y\). Each component plays a role:
  • The number 2 is a constant coefficient, which a constant multiplying the variables.
  • The variables \(x\) and \(y\) can vary, representing different values in each situation.
Algebraic expressions can be used to model real-world problems, help in solving equations, and simplifying mathematical problems. They provide a way to compactly describe relationships between numbers and variables.
Distributive Property
The distributive property is a fundamental concept in algebra that allows you to multiply a single term by terms within a parenthesis. It's expressed as \(a(b + c) = ab + ac\). This law helps simplify expressions and solve equations involving multiplication.
In the exercise provided, you encounter the distributive property when simplifying \((2x)(2y)\). Here, instead of a written sum, we distribute each multiple directly:
  • Multiply the constants: \(2 \times 2 = 4\).
  • Combine the variables: \(x \times y = xy\).
  • The result is \(4xy\).
This step demonstrates how the distributive property helps in systematically breaking down and simplifying expressions with multiple terms.
Mathematical Equivalence
Mathematical equivalence means two expressions represent the same quantity. If two expressions yield the same result for every value of their variables, they're equivalent. Evaluating equivalence involves simplifying expressions to their simplest form and comparing them.
In this particular context, we compare the simplified forms of \(2xy\) and \(4xy\). By doing so, we see:
  • The first expression is \(2xy\).
  • The second expression becomes \(4xy\) after simplification.
Since these expressions are not identical, they are not equivalent. Recognizing equivalence is crucial for solving problems correctly and ensuring expressions accurately model real-world scenarios or mathematical principles.

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