Problem 108 - Algebra A Combined Function

Question

Insert parentheses so that the following expression simplifies to 28 . $$ 2 \cdot 5+3^{2} $$

Step-by-Step Solution

Answer

Place parentheses as \(2 \cdot (5 + 3^{2})\) to get 28.

Step 1: Understand the Problem

We need to insert parentheses in the expression \(2 \cdot 5 + 3^{2}\) so that it simplifies to 28. Without parentheses, the expression evaluates as \(2 \cdot 5\) plus \(3^{2}\), which is 10 + 9 = 19. Our task is to find the right placement for parentheses to make the result 28.

Step 2: Analyze the Expression Components

The expression consists of two main parts: the multiplication \(2 \cdot 5\) and the addition of the square \(3^{2}\), which is already 9. We can try different combinations, prioritizing certain operations to change the result.

Step 3: Try Parentheses around Multiplication

Let's consider placing parentheses around \(2 \cdot (5 + 3^{2})\). Calculate inside the parentheses first: \(5 + 3^{2} = 5 + 9 = 14\). Then multiply: \(2 \cdot 14 = 28\).

Step 4: Confirm Solution

With parentheses placed as \(2 \cdot (5 + 3^{2})\), the expression evaluates to 28, which matches our required result. No further adjustments are needed.

Key Concepts

Parentheses in Mathematics

Parentheses are powerful tools in mathematics used to clarify the order of operations within an expression. They help dictate which parts of a mathematical expression should be calculated first. Without parentheses, expressions are typically evaluated according to a standard order of operations known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order ensures that calculations are performed consistently and accurately.

By inserting parentheses, we can override this standard order to ensure certain operations are completed first. This is essential when trying to manipulate expressions to achieve desired results, such as in the initial problem with the goal of simplifying to 28.

When parentheses are added to an expression, always compute the operations inside them first before moving on to the rest of the equation. This often changes the entire outcome of the expression, demonstrating how strategic use of parentheses can influence mathematical calculations dramatically.

Evaluating Expressions

Evaluating expressions is the process of calculating the mathematical value of an expression. This involves following a set of rules to simplify the expression and find its numerical result. In our example, the expression without any modifications is \(2 \cdot 5 + 3^2\).

Initially, according to the order of operations, we perform the multiplication and exponentiation before proceeding with the addition. This is because multiplication and exponentiation have a higher precedence over addition. Therefore, the expression evaluates as follows:

• First, compute the multiplication: \(2 \cdot 5 = 10\)
• Then the exponentiation: \(3^2 = 9\)
• And finally, the addition: \(10 + 9 = 19\)

However, to achieve the desired outcome of 28, we must rethink this process by rearranging the operations using parentheses.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves rewriting them in a simpler or more efficient form. This process makes them easier to manage and understand, particularly when solving math problems or real-world scenarios.

The original problem required redefining the expression to equal 28, which was impossible without altering the sequence of operations. To simplify effectively, we utilized parentheses strategically to adjust the computation process. By enclosing \(5 + 3^2\), we ensured that this sum would be calculated first:

• Inside the parentheses: \(5 + 9 = 14\)
• Then, multiply by 2: \(2 \cdot 14 = 28\)

This demonstrates how rearranging elements and simplifying can lead to a desired and correct outcome. Simplification often involves grouping terms and carefully selecting operations, balancing efficiency with correctness to achieve specific goals.