Problem 54 - Prealgebra

Question

Use the rule for order of operations along with the rules for addition, subtraction, and multiplication to simplify each of the following expressions. $$4(-3)(2-5)$$

Step-by-Step Solution

Answer

The expression simplifies to 36.

Step 1: Parentheses First

We start by simplifying the expression inside the parentheses. The expression inside the parentheses is \(2 - 5\). Calculate this to get \(-3\). So the expression becomes \(4(-3)(-3)\).

Step 2: Multiply the First Two Numbers

Next, we multiply the numbers outside of any parentheses from left to right. Start with \(4\) and \(-3\) to get \(-12\). Now, the expression simplifies to \(-12(-3)\).

Step 3: Multiply the Resulting Numbers

Finally, multiply \(-12\) by \(-3\). The product of two negative numbers is positive. Therefore, \(-12 \times -3 = 36\).

Key Concepts

Simplifying Expressions

When tackling mathematical problems, simplifying expressions is a nifty way to make complex problems more understandable. Simplification involves breaking the problem into smaller, more manageable parts until you're left with an easy-to-solve equation. This process can include removing parentheses, combining like terms, and performing operations that make the expression simpler. In our example expression:

• Start by identifying parts that can be simplified immediately via operations, such as the expression inside the parentheses.
• Follow the order of operations to ensure each part of the expression is handled in the correct sequence.
• Apply mathematical rules such as multiplication or addition where applicable.

This approach turns a potentially daunting problem into one that's easier to solve step-by-step.

Parentheses

Parentheses are used in mathematical problems to prioritize operations, ensuring some calculations are completed before others, in accordance with the order of operations.

• In our expression, \(4(-3)(2-5)\), the parentheses around \(2-5\) indicate that this segment should be simplified first before moving on to other operations.
• By solving \(2-5\) first, we transform that part of the expression to \(-3\), simplifying the entire expression to \(4(-3)(-3)\).

Using parentheses wisely can greatly impact your ability to solve math problems quickly and correctly. Don't rush this step and take your time to ensure the sequence of operations is preserved.

Multiplication Rules

Understanding multiplication rules is essential for simplifying expressions, especially when dealing with multiple terms. Two key rules related to multiplication are:

• The product of two positive numbers is positive.
• The product of two negative numbers is also positive, which means \(- imes - = +\).

In this particular problem, once the expression has been simplified within the parentheses, you multiply consecutively from left to right.
Start by calculating \(4 imes -3\), which equals \(-12\). Then, solve \(-12 imes -3\) using our rule about negative numbers, arriving at the positive \(+36\). This demonstrates the importance of multiplication rules when simplifying expressions containing negative values.

Addition and Subtraction Rules

Before diving into multiplying terms, ensure you have a solid grasp of how addition and subtraction rules come into play in expressions.
These operations are generally simpler and set the stage for more complex operations like multiplication and division. Here are a few guidelines:

• Both addition and subtraction are handled as per the standard order of operations, meaning they are performed after operations inside parentheses, exponents, and multiplication/division.
• Look at the sign between numbers; a minus is actually adding a negative. So \(2-5\) is truly \(2 + (-5)\), simplifying finally to \(-3\).

Applying these rules ensures that the foundational arithmetic components of any expression are correctly managed, paving the way for accurate simplification of recurring mathematical expressions.