Problem 33 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(2[5+2(8-3)]\)

Step-by-Step Solution

Answer

The simplified expression is 30.

Step 1: Evaluate the expression inside the innermost parentheses

The innermost expression is \((8-3)\). Subtract 3 from 8 to get 5. So, the expression becomes \(2[5+2(5)]\).

Step 2: Multiply within the brackets

Now we have \(2(5)\) inside the brackets. Multiply 2 by 5 to get 10. The expression is now \(2[5+10]\).

Step 3: Add within the brackets

Add the numbers inside the brackets: \(5+10 = 15\). This updates the expression to \(2[15]\).

Step 4: Simplify the final multiplication

Multiply 2 by 15 to simplify: \(2 \times 15 = 30\). Therefore, the expression simplifies to 30.

Key Concepts

Simplifying Expressions

Simplifying expressions involves performing operations in a way that reduces an expression down to its simplest form. The process of simplification usually involves combining like terms and performing basic arithmetic operations until you can't simplify any further.

In the case of our example expression, we follow specific steps to simplify:

• Start by solving expressions within parentheses or brackets to deal with smaller chunks first.
• Perform arithmetic operations such as addition, subtraction, multiplication, or division.
• Continue applying these operations following the order of operations (PEMDAS/BODMAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

The goal is to reach a single numerical answer, reducing complexity and making the expression easier to read and understand.

Parentheses

Parentheses play a crucial role in mathematical expressions. They dictate the order in which operations should be performed, ensuring that parts inside the parentheses are calculated before anything else outside them.

Think of parentheses as instructions to "do this part first." In the provided exercise, we see \( 8 - 3 \).

• This calculation takes precedence over operations that come after.
• Removing the parentheses without solving them first would lead to incorrect outcomes.

Parentheses can sometimes be nested, meaning one set of parentheses within another, as seen in our original expression \( 2[5 + 2(8 - 3)] \). The innermost parentheses are calculated first, and their result affects the external calculations. This ensures a structured approach to solving complex expressions efficiently.

Arithmetic Operations

Arithmetic operations are the building blocks of mathematics. They include addition, subtraction, multiplication, and division. Each operation has specific rules that dictate how numbers interact with each other. Understanding these operations helps in simplifying expressions like the example provided.

Here's a quick guide to arithmetic operations:

• Addition: Combines numbers into a total sum.
• Subtraction: Finds the difference between numbers by taking one away from another.
• Multiplication: Repeatedly adds a number to itself, resulting in a product.
• Division: Splits a number into equal parts or groups.

In the original problem, we used subtraction first to solve \( 8 - 3 \), then carried out multiplication within the brackets, followed by an addition, and finally multiplied again to reach the final solution.