Problem 35 - 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. $$-3+2(5-3)$$

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Parentheses

In the expression \(-3 + 2(5 - 3)\), we first address the parentheses. Calculate the result of \(5 - 3\) which equals \(2\). This simplifies the expression to \(-3 + 2 \times 2\).

Step 2: Multiplication

Now that the expression is simplified to \(-3 + 2 \times 2\), perform the multiplication operation first. Calculate \(2 \times 2\) which equals \(4\). The expression now becomes \(-3 + 4\).

Step 3: Addition/Subtraction

Finally, evaluate the remaining operation in the expression \(-3 + 4\). This is equivalent to \(4 - 3\), resulting in \(1\). Thus, the simplified expression evaluates to \(1\).

Key Concepts

Understanding Parentheses in Order of Operations

Parentheses play a crucial role in mathematical expressions because they dictate which parts of the expression should be calculated first. In the order of operations, which is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), parentheses come first.

This means that any calculation inside parentheses must be completed before moving on to other operations. For example, in the expression \(-3 + 2(5 - 3)\), you first focus on solving \(5 - 3\) inside the parentheses. This results in \(2\), simplifying the expression to \(-3 + 2 \times 2\). Remembering to tackle parentheses first helps avoid mistakes and ensures the correct result.

By addressing what's inside the parentheses early, you set the stage for a smooth and accurate simplification of the entire expression.

Multiplication: The Next Step After Parentheses

After solving the operations inside the parentheses, the next step according to the order of operations is multiplication. In our simplified expression \(-3 + 2 \times 2\), multiplication must be tackled before any addition or subtraction.

Here, you have the operation \(2 \times 2\), which simplifies to \(4\). It's important to compute this result before moving on to any other arithmetic operations, as multiplication takes precedence over addition and subtraction.

So, remember after resolving any parentheses, always shift your focus to multiplication and division next. By following this rule, you ensure that your calculations remain consistent with the order of operations, avoiding common pitfalls and errors.

Addition and Subtraction: Finalizing the Expression

Once you've dealt with parentheses and performed any necessary multiplication or division, you move on to the final steps: addition and subtraction. In the expression \(-3 + 4\) (from our previous simplifications), you simply need to complete the arithmetic from left to right.

Unlike multiplication and division, addition and subtraction are performed in the order they appear from left to right, rather than having priority over one another.

In this case, \(-3 + 4\) simplifies straightforwardly to \(1\), completing the problem.

• Perform addition and subtraction as they occur from left to right.
• Ensure all previous steps with parentheses, multiplication, and division are already addressed.

Following this final step correctly will lead you to the accurate simplification of any arithmetic expression.