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Chapter 1: Problem 18

Simplify. $$ 4 \cdot 3 \div 12 \cdot 2-(-2) 2 $$

Short Answer

Expert verified
The simplified expression is -2.

Step by step solution

01

Identify Operations

The expression given is \(4 \cdot 3 \div 12 \cdot 2 - (-2) \times 2\). It includes multiplication, division, and subtraction operations. Multiplication and division need to be performed before subtraction according to the order of operations.
02

Perform Multiplication and Division Left to Right

We perform the multiplication and division from left to right. Start with:1. \(4 \cdot 3 = 12\)2. \(12 \div 12 = 1\)3. \(1 \cdot 2 = 2\)This simplifies the left portion of the expression to \(2\).
03

Simplify Negative Multiplication

Simplify the negative exponent and its multiplication: \(-(-2) \times 2\). The double negative \(-(-2)\) becomes positively \(2\), so we have:\(2 \times 2 = 4\).
04

Subtract the Results

Now, substitute these simplified values back into the expression:\(2 - 4\). Performing the subtraction gives \(2 - 4 = -2\).

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Key Concepts

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

Order of Operations
Understanding the order of operations is crucial in algebraic simplification. To remember the correct order, use the acronym PEMDAS:
  • Parentheses
  • Exponents
  • Multiplication and Division
  • Addition and Subtraction
It's essential to follow this order to solve expressions correctly. In the given expression, the first focus is on multiplication and division. These operations take precedence over subtraction. Working from left to right for both multiplication and division ensures the expression simplifies correctly. Without proper order, solutions won't be accurate. By consistently applying PEMDAS, you maintain the logical flow required in algebraic calculations.
Multiplication and Division
In algebra, multiplication and division appear frequently. They have equal standing in the order of operations. This means they are performed based on their left-to-right appearance in the expression. Let鈥檚 break it down:When simplifying the expression
  • Start with the multiplication: \(4 \cdot 3 = 12\).
  • Follow with division: \(12 \div 12 = 1\).
  • Continue with multiplication: \(1 \cdot 2 = 2\).
By following each step accurately, you ensure consistency and correctness. Skipping the left-to-right approach or confusing multiplication with addition changes the final outcome.
Negative Numbers
Working with negative numbers can sometimes be tricky. It's important to handle signs correctly to get accurate results.In our expression, the tricky part was \(-(-2) \times 2\):
  • The double negative \(-(-2)\) turns into a positive. This yields \(2\).
  • Multiply this result: \(2 \times 2 = 4\).
Understanding how negative signs interact is key. A single error with negatives can lead to incorrect answers. It's always good to double-check your work to ensure signs are managed properly, especially when they appear next to each other.

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