Problem 51 - 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. $$-3(4-7)-2(-3-2)$$

Step-by-Step Solution

Answer

The expression simplifies to 19.

Step 1: Simplify Inside the Parentheses

First, simplify the expressions inside the parentheses. For the expression \(4-7\), calculate: \(4-7 = -3\). Then, for the expression \(-3-2\), calculate: \(-3-2 = -5\). This changes the original expression to: \[-3(-3)-2(-5)\]

Step 2: Multiply Inside Terms

Next, multiply the numbers outside the parentheses with the simplified values inside them. For \(-3(-3)\), multiply the numbers: \(-3 imes -3 = 9\). Similarly, for \(-2(-5)\), multiply the numbers: \(-2 imes -5 = 10\). Now, the expression becomes: \[9 + 10\]

Step 3: Add the Results

Finally, add the products obtained from step 2: \(9 + 10 = 19\). This is the simplified result of the expression.

Key Concepts

Simplification

Simplification is all about making math expressions easier to work with. You break them down into smaller, manageable parts to find the answer easily. When simplifying, it's crucial to follow a set order: addressing operations like parentheses first, then multiplication and division, and finally, addition and subtraction. This process helps prevent mistakes and ensures you get the right result.
Simplifying involves:_

• Breaking down complex expressions into simpler parts
• Following the order of operations correctly
• Avoiding errors that could arise from incorrect calculations_

When working on a math problem, keep the goal of simplification in mind to make computations as straightforward and painless as possible.

Parentheses

Parentheses change the order in which you handle parts of a mathematical expression. They signal which operations you need to perform first. If you see parentheses, always tackle the calculations inside them before moving on to other operations.
In the exercise, this principle is used by solving the operations within parentheses _before_ anything else:

• Calculate the result of expressions inside parentheses first, such as from \(4-7\) resulting in \(-3\)
• Find the solution to \(-3-2\), which yields \(-5\)

Once you handle what's inside the parentheses, you can then proceed with other operations in the correct sequence. This keeps your work organized and prevents missteps.

Multiplication Rules

Multiplication rules are essential when working with expressions that have multiple steps, like dealing with parentheses. Once the expressions inside parentheses are simplified, often you need to multiply the results with the numbers outside the parentheses.

• Identify which numbers are to be multiplied, like \(-3\) times \(-3\) in the given expression, resulting in a positive \(9\) due to the negative signs cancelling
• Similarly, perform \(-2\) times \(-5\), leading to the positive product \(10\)

The key points to remember with multiplication:

• Two negative numbers multiply to give a positive number
• Proceed systematically to multiply, ensuring you don't skip any steps

With these multiplication rules in place, your calculations will be consistent and reliable.

Addition and Subtraction Rules

Addition and subtraction are the final steps in simplification after other operations are complete. They follow the multiplication and division operations, meaning they come last.
In this exercise, after multiplying to get \(9\) and \(10\), you simply sum these products:

• Add \(9\) to \(10\) resulting in \(19\)

This step is about combining all parts into a single simplified result:

• Make sure to carefully add or subtract the outcomes of previous calculations
• Double-check your work to avoid small mistakes that can occur during this final step

By mastering addition and subtraction rules, you bring your work to a satisfying and accurate conclusion.