Problem 98 - Algebra A Combined Function

Question

Are parentheses necessary in the expression \((2+3) \cdot 5 ?\) Explain your answer.

Step-by-Step Solution

Answer

Yes, parentheses are necessary to ensure the correct calculation.

Step 1: Understanding the Expression

First, evaluate the expression with the parentheses. The expression given is \((2 + 3) \cdot 5\). According to the order of operations, we first perform the operation inside the parentheses. Thus, \(2 + 3 = 5\). This simplifies the expression to \(5 \cdot 5\).

Step 2: Evaluate Without Parentheses

Now, let's evaluate the expression as if the parentheses were not there: \(2 + 3 \cdot 5\). According to the order of operations (PEMDAS/BODMAS), we perform multiplication before addition. Hence, first, calculate \(3 \cdot 5 = 15\). Then, add \(2\) to \(15\), resulting in \(17\).

Step 3: Compare Results

With parentheses, \((2 + 3) \cdot 5\) evaluates to \(25\). Without parentheses, the expression \(2 + 3 \cdot 5\) evaluates to \(17\). These results are different, indicating parentheses change the outcome of the expression.

Key Concepts

Parentheses in Math

Parentheses in math are like powerful tools that determine the flow of an expression or equation. They tell us which operations to perform first before anything else. Using parentheses is crucial when you want certain calculations to be prioritized over others.
When you see an expression like \[(2 + 3) \cdot 5,\]you are immediately guided to do the addition inside the parentheses first. This results in \(2 + 3 = 5\), then the new expression becomes \[5 \cdot 5 = 25.\]Thus, parentheses ensure that calculations are not only organized but also give you the intended result.
Without parentheses, the natural flow dictated by operation order might be lost, potentially leading to wrong results. Hence, understanding parentheses helps you control and navigate expressions effectively.

PEMDAS

PEMDAS is a helpful acronym that captures the essence of order of operations in mathematics. It stands for:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

These operations are performed from left to right, but following the hierarchy dictated by PEMDAS ensures that expressions are evaluated correctly.
Consider the example from earlier: \[2 + 3 \cdot 5.\]Since there are no parentheses guiding us otherwise, PEMDAS tells us to do multiplication before addition. Therefore, we compute \[3 \cdot 5 = 15 \]first, and then \[2 + 15 = 17.\]This structured method reduces confusion and miscalculations, allowing for precise and clear mathematical computations.

BODMAS

BODMAS is another acronym that serves the same purpose as PEMDAS in regions like the UK. It stands for:

• B: Brackets
• O: Orders (i.e., powers and roots, such as squares and square roots)
• D: Division
• M: Multiplication
• A: Addition
• S: Subtraction

Similar to PEMDAS, BODMAS guides the order in which mathematical operations should be performed. Brackets are the first point of interest, ensuring that any expressions within them are handled first.
For the expression without parentheses:\[2 + 3 \cdot 5,\]using BODMAS, you still perform the multiplication before the addition. So, \[3 \cdot 5 = 15,\]and then the final addition becomes:\[2 + 15 = 17.\]Whether you use PEMDAS or BODMAS, both methods highlight the necessity of understanding operation order for accurate mathematical problem-solving.