Problem 91 - Prealgebra

Question

Use the rule for order of operations to combine the following. \((7-3)(8+2)\)

Step-by-Step Solution

Answer

The expression evaluates to 40.

Step 1: Simplify Inside Parentheses

First, we need to simplify the expressions within each set of parentheses. For the expression \((7-3)\), subtract 3 from 7, which equals 4. For the expression \((8+2)\), add 8 and 2, which equals 10. So, the expression becomes \(4 \times 10\).

Step 2: Multiply the Results

Now, apply the multiplication rule. Multiply the numbers obtained from the simplified parentheses: \(4 \times 10 = 40\).

Key Concepts

Parentheses

Parentheses are an essential part of mathematical expressions and help dictate the correct order of operations. When you see expressions enclosed in parentheses, solve whatever is inside them first. By doing this, you ensure that you follow the operations in the correct sequence as outlined by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). For instance, consider an expression

• Within \((7-3)\), subtract 3 from 7, resulting in 4.
• Within \((8+2)\), add 8 and 2, resulting in 10.

Once simplified, the expression now looks cleaner and becomes easier to approach in the next steps.

Multiplication

Multiplication comes after dealing with operations inside parentheses. Once any expressions within parentheses are simplified—and assuming there are no exponents—you proceed with multiplication. Recall our example, where after simplifying parentheses, you are left with \(4 \times 10\).

• Multiply these two numbers: \(4 \times 10 = 40\).

This key step shows the importance of executing operations in the correct order, as even a small mistake could lead to an entirely different result. Always verify your multiplications to ensure accuracy as this operation forms a pivotal step in solving expressions correctly.

Simplify Expressions

Simplifying expressions involves combining like terms and performing operations in the set sequence. The process requires careful execution of each step—starting with parentheses, followed by multiplication, and continuing as needed based on the expression's complexity. This disciplined method ensures that the expressions are reduced to their simplest form.

• Breaking it down: start by handling all operations within the parentheses.
• Proceed to execute any multiplications.
• By doing so, the original complicated expression simplifies, in this case, to a single number, \(40\).

Simplification is all about making expressions easier and cleaner to understand, ensuring that they convey results in the clearest and most concise form possible. By following these steps diligently, you gain confidence in managing more complex mathematical problems through a structured approach.