Problem 100 - Elementary Algebra Within Reach

Question

In Exercises 97-102, evaluate the expression. $$ 6+3(4+2) $$

Step-by-Step Solution

Answer

The evaluated expression value is 24.

Step 1: Evaluate Operation inside Parentheses

Start by performing the operation inside the parentheses:\(4+2 = 6\). Now replace the parentheses in the original equation with the result, the equation becomes: \(6 + 3*6\).

Step 2: Perform Multiplication

Next, perform the multiplication operation: \(3*6 = 18\). Replace the multiplication operation in the equation from Step 1 with the result, the equation becomes: \(6 + 18\).

Step 3: Perform Addition

Lastly, perform the addition operation: \(6 + 18 = 24\).

Key Concepts

Parentheses

When you encounter parentheses in a mathematical expression, they serve as an instruction to perform the operations inside them first. This is a key component of the Order of Operations, which helps ensure that everyone solves problems the same way. Think of parentheses as the top priority in the step-by-step order of solving expressions. In cases where you have more than one set of parentheses, you would evaluate the inner parentheses first, moving outward.

For example, inside the expression from our exercise, we have:

• The expression inside the parentheses is \(4+2\).
• We first calculate this sum to get \(6\).

Once you have dealt with the contents of the parentheses, you can proceed to the next steps in the expression, focusing on whatever operations remain.

Multiplication

After resolving any parentheses, multiplication comes next in the Order of Operations. It is performed before addition or subtraction when no parentheses dictate otherwise. The importance of handling multiplication early is to ensure that larger sums or differences are not prematurely calculated. This ensures accuracy and consistency across various mathematical problems.

In our example:

• With the result of the parentheses being \(6\), the expression becomes \(6 + 3 \times 6\).
• We then perform the multiplication: \(3 \times 6 = 18\).

After multiplication, the expression is simplified further before proceeding to any addition or subtraction.

Addition

Addition is typically one of the final steps when solving expressions, according to the Order of Operations, unless you're inside a set of parentheses. After addressing all parentheses and executing all multiplication or division (as applicable), you conclude with addition. By following this order, it ensures that you create the simplest version of the equation ready for the sum.

In our ongoing example:

• Once multiplication has been addressed, the remaining task is to add: \(6 + 18\).
• This yields the final result of \(24\).

Understanding the proper placement of addition in this structure reduces errors and makes complex operations more manageable.