Problem 31 - 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. $$4(-3+2)$$

Step-by-Step Solution

Answer

The simplified expression is \(-4\).

Step 1: Identify Operations Inside Parentheses

First, look for any operations that need to be done inside parentheses. In the expression \(4(-3+2)\), the operation inside the parentheses is \(-3 + 2\).

Step 2: Simplify Inside Parentheses

Now, perform the addition inside the parentheses: \(-3 + 2 = -1\). The expression now becomes \(4(-1)\).

Step 3: Multiply

Next, multiply the number outside the parentheses by the result inside the parentheses. Multiply \(4\) by \(-1\), which gives \(4 \times (-1) = -4\).

Step 4: Finalize the Expression

The simplified expression is \(-4\).

Key Concepts

Understanding Parentheses

In mathematics, parentheses play a vital role when evaluating expressions and ensuring computations are carried out correctly. They signal the need to perform the operation within them first, according to the order of operations.

This order, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates that any operation inside parentheses should be completed before proceeding with other calculations. For example, in the expression \(4(-3+2)\), the calculation inside the parentheses, \(-3 + 2\), should be addressed first before multiplying by 4.

Dealing with parentheses helps in managing more complex math problems, breaking them into smaller, more manageable tasks. Always prioritize solving these inner calculations to evaluate the expression correctly.

The Basics of Addition

Addition is one of the four elementary operations of arithmetic, denoted by the plus sign \(+\). It is the process of calculating the total of two or more numbers or amounts. In our example, within the parentheses \(-3 + 2\), we add the numbers -3 and 2 together.

Here's how it works:

• Negative numbers decrease the value; thus, adding -3 to 2 results in a smaller value.
• When adding opposite signs, subtract the smaller number from the larger, and take the sign of the larger number.
• In this case, \(-3 + 2\) equals -1 because 3 is larger and negative.

Understanding such simple additions is crucial for simplifying expressions. Once inside-parentheses calculations are done, we can move forward with other operations.

Steps of Multiplication

Multiplication is another key arithmetic operation, symbolized by the multiplication sign \(\times\). It is essentially repeated addition. When we multiply, we are adding a number to itself a specified number of times.

For example, in the expression \(4(-1)\), we multiply 4 by -1, which means adding four zeros of -1, resulting in \(-4\). Here's a breakdown:

• Identify the terms to be multiplied: \(4\) and \(-1\).
• Recall that multiplying a positive number by a negative number results in a negative product.
• Perform the multiplication: \(4 \times (-1) = -4\).

Mastering multiplication and understanding its interaction with negatives is crucial for progressing in math. This concept extends to many branches, including algebra and calculus, providing the groundwork for more advanced calculations.