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Chapter 5: Problem 108

Simplify. (Assume variables are nonzero.) $$ (2 x 2 y 3) 0 $$

Short Answer

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Step by step solution

01

Understand the Expression

The expression given is \((2 \, x \, 2 \, y \, 3) \times 0\). This means we have a product of several factors multiplied by zero.
02

Apply Zero Property of Multiplication

The Zero Property of Multiplication states that any number multiplied by zero is zero. This applies to the entire expression since everything is being multiplied together with zero.
03

Final Simplified Result

Since the product of any expression with zero is zero, the entire expression simplifies to 0.

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Key Concepts

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

Simplifying Expressions
Simplifying expressions means breaking down mathematical expressions to their simplest form. It involves performing all possible operations so that the expression cannot be simplified further.
In the given exercise, we started with a complex expression involving numbers and variables, specifically \(2 \, x \, 2 \, y \, 3\).
  • Identify and apply any relevant mathematical properties.
  • Combine like terms or perform arithmetic calculations if possible.
  • Incorporate constant factors or use specific numerical properties.

Using the Zero Property of Multiplication, any attempt to further treat or rearrange the expression stopped because multiplying by zero inevitably led to zero as the simplest form.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operations like addition or multiplication.
In the example \(2 \, x \, 2 \, y \, 3\), each letter represents a variable that stands for some unknown number.
  • Variables are symbols in an algebraic expression that can take different values.
  • An expression can combine variables with numbers and operate on them.
  • When multiplying, you simply multiply numeric coefficients and consider the operation of variables separately.

Understanding algebraic expressions means recognizing how the components interact, such as through operations like multiplication as highlighted in our exercise.
Despite the complexities, the Zero Property simplifies outcomes dramatically when zero is involved.
Properties of Multiplication
The Properties of Multiplication are rules that help manipulate and simplify expressions involving multiplication.
Several key properties make it easier to comprehend how numbers and variables interact in multiplication:
  • Commutative Property: The order of multiplication does not change the result. For example, \(a \, \times \, b = b \, \times \, a\).
  • Associative Property: The way grouped numbers are multiplied doesn't affect the outcome. For instance, \( (a \, \times \, b) \, \times \, c = a \, \times \, (b \, \times \, c)\).
  • Distributive Property: You can multiply a single term and a sum, \(a(b+c) = ab + ac\).
In the exercise, the Zero Property was crucial. This property asserts that any number multiplied by zero results in zero.
This helps simplify expressions easily when zero is a factor, reinforcing the straightforward result our expression simplifies to 0.

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