Problem 4 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. \(10+2 \times(5-3)\)

Step-by-Step Solution

Answer

The simplified expression is 14.

Step 1: Solve Inside the Parentheses

Begin by solving the expression inside the parentheses: \(5 - 3\). This simplifies to \(2\).

Step 2: Substitute and Simplify

Substitute \(2\) back into the expression, replacing \((5-3)\), giving us: \(10 + 2 \times 2\).

Step 3: Apply the Order of Operations

Next, follow the order of operations (PEMDAS/BODMAS), starting with multiplication: \(2 \times 2 = 4\). So, the expression becomes \(10 + 4\).

Step 4: Final Addition

Finally, add the remaining numbers: \(10 + 4 = 14\). Thus, the simplified expression is \(14\).

Key Concepts

Understanding Parentheses

Parentheses are an important part of mathematical expressions and they signal the need for certain operations to be performed first. Think of parentheses like a spotlight on a stage, highlighting the part of the expression that requires immediate attention. They tell you which calculations you must complete before moving on to other operations. For instance, in the expression \(10 + 2 \times (5-3)\), the operation within the parentheses, \(5-3\), should be solved first. Treating what's inside the parentheses as a priority ensures that the rest of the expression is calculated correctly.
When you see parentheses in expressions, always deal with them first. This rule is based on the broader set of guidelines known as the order of operations.

Diving into Multiplication

Multiplication is one of the fundamental operations in mathematics, symbolized often by \( \times \) or simply placing numbers next to each other, like \(ab\). Once parentheses are resolved, multiplication typically comes next in line, according to the order of operations rule, known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
In the problem \(10 + 2 \times 2\), after solving the parentheses, we move on to multiplication before addressing addition. The multiplication of \(2 \times 2\) gives \(4\), which is crucial to solving the expression. This operation changes the expression significantly and helps bring clarity to the remaining steps.

Exploring Addition

Addition is often the final step in simplifying expressions once all other operations following the order of operations have been completed. The result of an addition operation is the sum, which provides the total value when two or more numbers are combined. In our provided example, after parentheses and multiplication have been processed, the last operation is taking \(10\) and adding it to \(4\).
With the calculation \(10 + 4\), you simply add the two numbers to find the sum, which is \(14\). This clean finish emphasizes how simplification techniques, like the order of operations, work together to convert complex expressions into simple solutions.