Problem 97 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

Parentheses are not necessary; they do not change the expression's result.

Step 1: Understanding Order of Operations

According to the order of operations in mathematics, also known as PEMDAS/BODMAS, operations within parentheses are performed first. This means that any operation inside parentheses gets priority in calculations.

Step 2: Evaluating the Given Expression with Parentheses

In the expression \(2+(3 \cdot 5)\), the operation inside the parentheses is \(3 \cdot 5\), an example of multiplication. According to the order of operations, it will be calculated first regardless of the parentheses. Thus, the expression simplifies sequentially as follows: \(2 + (3 \cdot 5) = 2 + 15 = 17\).

Step 3: Evaluating the Expression Without Parentheses

Without the parentheses, the expression becomes \(2 + 3 \cdot 5\). Following the order of operations, multiplication should be done before addition. Therefore, the expression simplifies the same way: \(2 + 3 \cdot 5 = 2 + 15 = 17\).

Step 4: Conclusion on the Necessity of Parentheses

Since the result of the expression is the same with or without the parentheses, the parentheses are not strictly necessary in this case. The multiplication operation will be prioritized in both cases, leading to the same outcome.

Key Concepts

Parentheses

When dealing with mathematical expressions, parentheses play an essential role in guiding the order in which calculations are performed. Think of them as signals that tell you to do certain operations first before anything else.
The basic function of parentheses is to group certain parts of an expression to make sure those parts are calculated first. This grouping can change the outcome of an expression if not considered properly.

• For example, in the expression \((2 + 3) \cdot 5\), the sum \(2 + 3\) within parentheses calculates to 5, making the entire expression equal \(5 \cdot 5 = 25\).
• Contrastingly, without parentheses, the expression \(2 + 3 \cdot 5\) would be calculated as \(2 + (3 \cdot 5) = 2 + 15 = 17\), which yields a different result.

In essence, parentheses are crucial when the intended order of operations is different from what standard mathematical conventions dictate. It’s a simple yet powerful tool to master in math.

Mathematical Expressions

Mathematical expressions are combinations of numbers, operations, and sometimes variables that represent a particular value. Understanding how to interpret these expressions is a foundational skill in math.
Expressions can range from basic arithmetic, like \(5 + 3\), to more complex types, including algebraic expressions like \(2x + 3\).
What makes expressions particularly interesting is the need to evaluate them correctly, adhering to mathematical conventions such as the order of operations.

• Expressions are often simplified by reducing them step by step, paying careful attention to the order of operations.
• Ensuring clarity in expressions through proper notation, like parentheses, prevents misunderstandings in calculations.

Clear comprehension of mathematical expressions is necessary to accurately tackle more complicated formulas and theorems in math.

Multiplication Before Addition

One of the key rules in evaluating mathematical expressions is that multiplication is done before addition. This follows a widely accepted convention known by the acronym PEMDAS/BODMAS, which stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.
Whenever you encounter an expression involving both multiplication and addition, you should first perform all multiplication operations before moving on to addition.

• Take the expression \(2 + 3 \cdot 5\) as an example. Here, you first calculate \(3 \cdot 5 = 15\), then add 2 to get the final result of 17.
• This rule is not merely a suggestion but a standard that ensures consistent results across all computations.

Exceptions come when parentheses dictate a different calculation order. However, when there are no parentheses, always remember to multiply before you add! This recognition and adherence will help you avoid common calculation errors.