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Chapter 2: Problem 73

Simplify each expression. $$0(4 z)$$

Short Answer

Expert verified
The expression simplifies to 0.

Step by step solution

01

Recognize the Zero Product Rule

In any multiplication problem, if any term in the multiplication is zero, the product will always be zero. This is known as the zero product property.
02

Apply the Rule to the Given Expression

Identify that the expression is a multiplication of 0 and another term, which is \(0(4z)\). Since 0 is being multiplied by \(4z\), use the zero property to determine that the product is 0.
03

Write the Simplified Expression

After applying the zero product rule, the simplified expression is simply 0.

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Key Concepts

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

Zero Product Rule
The Zero Product Rule is a fundamental principle in algebra that can simplify your calculations neatly. This rule states that if the product of two or more numbers is zero, then at least one of the numbers must be zero. This property is particularly useful for solving equations and simplifying expressions involving zero.
For example, in the expression \(0(4z)\), the rule dictates that because one of the factors is zero, the entire product is zero. This avoids the need for lengthy calculations; you quickly and efficiently know the entire product without the need to multiply all factors.
Simplifying Expressions
Simplifying expressions involves reducing them to their most compact form while maintaining the same value. This process helps make the equations or expressions easier to understand and solve.
In prealgebra, simplifying an expression often includes:
  • Combining like terms.
  • Applying arithmetic operations correctly.
  • Using properties like the Zero Product Rule.
In the exercise \(0(4z)\), simplifying it involves recognizing the multiplication by zero, hence the expression simplifies directly to zero. This act of simplification fosters a deeper understanding of algebraic principles and the elegance of mathematical shortcuts.
Multiplication
Multiplication is one of the basic arithmetic operations, crucial in mathematics. It involves calculating the total of one number added repeatedly the number of times indicated by another number.
Here are some key points to understand about multiplication:
  • It is commutative, meaning that \(a \times b = b \times a\).
  • It is associative, so \((a \times b) \times c = a \times (b \times c)\).
  • Involves repeated addition, for instance, \(3 \times 4 = 4 + 4 + 4 = 12\).
  • Any number multiplied by zero results in zero due to the Zero Product Rule.
By correctly applying multiplication principles, large calculations can be broken down, streamlining complex mathematical tasks. In problems like \(0(4z)\), understanding that multiplying by zero guarantees a simple zero output underscores the power of multiplication simplicity.

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