Problem 81 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ -3(-4)(0) $$

Step-by-Step Solution

Answer

The result of the operation is 0.

Step 1: Understanding the Problem

We are asked to perform the operations on the expression \[-3(-4)(0)\].This involves understanding the multiplication of three numbers.

Step 2: Multiplying Two Negative Numbers

Start by multiplying the first two numbers: \(-3\) and \(-4\).When two negative numbers are multiplied, the result is positive:\(-3) \times (-4) = 12\).

Step 3: Multiplying by Zero

Now, take the result from Step 2, which is \(12\), and multiply it by zero:\(12 \times 0 = 0\).Any number multiplied by zero is always zero.

Step 4: Final Result

Since we've multiplied through all numbers, the final result of the expression \[-3(-4)(0)\] is \(0\).

Key Concepts

Negative Numbers Multiplication

Multiplying negative numbers can initially seem tricky, but it follows simple rules. When you multiply two negative numbers, the result is always positive. This is because negatives in multiplication actually cancel each other out. Think of it as reversing a reversal.

For example:

• If you multiply \(-3\) by \(-4\), you get \(12\). This is because \(-)\) multiplied by \((-)\) gives a positive. The two negatives act like opposing forces that turn the result positive.

Understanding these foundational rules of algebraic multiplication not only simplifies the process but also boosts confidence when handling more complex problems.

Multiplying by Zero

The rule of multiplying by zero is quite straightforward: any number multiplied by zero is always zero. This is a fundamental rule in mathematics, as it highlights the neutralizing effect of zero in multiplication.

For instance:

• If you have the result of a previous multiplication, such as \(12\) from multiplying \(-3\) and \(-4\), you simply multiply \(12\) by \(0\) to get zero.

It doesn’t matter how large or complex the numbers you've multiplied are; the moment you introduce a zero into the multiplication, the entire product becomes zero.

This rule is essential to remember for solving algebraic expressions efficiently.

Step-by-Step Problem Solving

Mastering a step-by-step approach to problem-solving is key in algebraic multiplication, as it helps break down potentially complex operations into clear, manageable parts.

Here’s how you can tackle problems systematically:

• Identify the Operation: Determine what kind of multiplication is required - are you dealing with negative numbers or zeros?
• Proceed in Stages: For multiple factors, multiply two numbers at a time to reduce complexity. As in our example, start with \(-3\) times \(-4\) to get \(12\).
• Apply Basic Rules: Know the basic rules, like zero multiplication leading to zero, to simplify the process.

By dissecting the problem into these steps, you not only reach the correct solution efficiently but also enhance your algebraic understanding and reasoning skills with practice.