Problem 43 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

Parentheses are not necessary in this expression as the order of operations dictates correct evaluation without them.

Step 1: Understanding the Expression

The given expression is \(2 + (3 \cdot 5)\). We see that there is an addition operator and a multiplication operator within parentheses.

Step 2: Applying Order of Operations

According to the order of operations (PEMDAS/BODMAS), operations inside parentheses must be done first. Therefore, the expression inside the parentheses, \(3 \cdot 5\), must be evaluated first.

Step 3: Evaluating Without Parentheses

If we remove the parentheses, the expression becomes \(2 + 3 \cdot 5\). Again, applying the order of operations, we must perform the multiplication before addition, resulting in the same calculation process.

Step 4: Conclusion on Parentheses

Since both \(2 + (3 \cdot 5)\) and \(2 + 3 \cdot 5\) would yield the same result (15), the parentheses in this specific expression are not necessary to ensure correct order of operations.

Key Concepts

Parentheses

Parentheses are symbols used in mathematics to group parts of an expression that need to be evaluated first. They help control the order in which operations are performed. In complex equations, parentheses indicate which operations are priority.

For example, in the expression \(2 + (3 \cdot 5)\), the parentheses tell us to calculate \(3 \cdot 5\) before adding \(2\). Without parentheses, we rely on standard operation rules.

Parentheses can simplify reading and understanding expressions. They clarify which operations should be done before others. If used incorrectly, they can drastically change the result of an equation.

PEMDAS

PEMDAS is an acronym that helps remember the order of operations in math. It stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This rule guides us through solving arithmetic expressions step by step.

• First, complete operations inside parentheses.
• Next, solve exponents.
• Then perform multiplication and division from left to right.
• Finally, solve addition and subtraction from left to right.

PEMDAS helps prevent errors in calculations. Just follow these rules, and you’ll safely simplify any arithmetic expression.

BODMAS

BODMAS is similar to PEMDAS. It stands for Brackets, Orders, Division and Multiplication, Addition and Subtraction. It helps remind us of the order to solve parts of mathematical expressions.

• Brackets: Solve expressions inside brackets first. "Brackets" is a term often used in countries like the UK instead of "parentheses."
• Orders: This step involves solving exponents or roots (like square roots).
• Division and Multiplication: Handle these from left to right.
• Addition and Subtraction: Perform these operations from left to right.

With BODMAS, you can confidently evaluate expressions just like with PEMDAS.

Arithmetic Expressions

An arithmetic expression is a combination of numbers, operators, and sometimes variables that express a computation. Expressions can include:

• Numbers (e.g., \(3, 5, 42\))
• Operators (e.g., \(+\), \(-\), \(\cdot\), \(/\))
• Parentheses (to group operations and determine priority)

Evaluating an arithmetic expression involves calculating its value by applying the correct operations in the right order. The order of operations, defined by PEMDAS/BODMAS, ensures expressions are simplified correctly. Understanding arithmetic expressions is key to solving math problems efficiently.