Problem 7 - Intermediate Algebra

Question

Fill in the blanks. In the expression \(9+6[22-(6-1)]\), the _____ are the innermost grouping symbols, and the brackets are the _____ grouping symbols.

Step-by-Step Solution

Answer

Parentheses, outer

Step 1: Identify Grouping Symbols

The expression given is \(9+6[22-(6-1)]\). Examining the expression, we can see different types of grouping symbols: parentheses \(()\) and brackets \([]\). The role of grouping symbols is to define the order in which parts of the expression are evaluated.

Step 2: Determine the Innermost Grouping Symbols

In the expression \(9+6[22-(6-1)]\), focus on the grouping symbols that appear at the deepest level. Here, the subtraction inside the parentheses \((6-1)\) must be evaluated first. Parentheses are the innermost grouping symbols used in this part of the expression.

Step 3: Identify the Outer Grouping Symbols

After evaluating the innermost grouping symbol, we observe the brackets \([ ]\) in \(6[22-(6-1)]\). Brackets are used to group everything together, indicating they are the next level of grouping.

Key Concepts

Parentheses

Parentheses \(()\) are fundamental grouping symbols in mathematics. They indicate that the operations inside them should be performed first. In expressions, parentheses can be considered as the starting point for solving problems.
For example, in the mathematical expression \(5 \times (3+2)\), the operation inside the parentheses \((3+2)\) is evaluated first, resulting in 5. Then, you multiply the result by 5, which equals 25.

Why Parentheses Matter

• Ensure correct sequence: Parentheses guide us in performing calculations in the right order, avoiding errors.
• Avoid ambiguity: They remove any confusion about which operations to perform first.

It's always important to observe the placement and use of parentheses when solving equations or expressions, as they play a crucial role in determining the order of operations, often acting as the innermost grouping symbols.

Brackets

Brackets \([ ]\) are also grouping symbols, but they usually come after parentheses in an expression. Brackets are used to further organize or prioritize calculations, especially in expressions that contain multiple layers of grouping symbols.
For example, in an expression like \([3 \times (2+4) + 7]\), the operations inside the parentheses are calculated first, resulting in \(6\). Then, the result is multiplied by 3 and added to 7, all within the brackets, ensuring these operations are completed before proceeding outside of the brackets.

Key Points About Brackets

• Level of hierarchy: Brackets are typically used after parentheses to create an additional layer of organization in complex expressions.
• Overall grouping: Sometimes, brackets serve to encapsulate larger parts of an expression, ensuring that particular operations are completed before others begin.

Brackets are essential when dealing with intricate problems or nested mathematical statements. They work hand-in-hand with parentheses to structure the sequence of operations accurately.

Order of Operations

The order of operations is a set of rules that dictates the sequence in which parts of a mathematical expression are evaluated. It prevents mix-ups and ensures consistency when solving complex expressions.
A common acronym to remember the order is PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This order helps to clarify which operations to perform first.

Importance of Order

• Logical sequence: Ensures accurate results by following a standardized sequence of operations.
• Consistency: Everyone arrives at the same answer when using a uniform process.

In our original problem \(9+6[22-(6-1)]\), following the order of operations is crucial. It determines that calculations inside the parentheses are done before the entire expression inside the brackets is evaluated, and then affects the operations outside of these grouping symbols. Adhering to the order of operations is key to solving mathematical expressions correctly.