Problem 33 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$6 \cdot 8 \div 4$$

Step-by-Step Solution

Answer

The simplified value of the expression \(6 \cdot 8 \div 4\) is \(12\).

Step 1: Identify the Operations

Identify the operations in the given expression. In this exercise, the operations are multiplication and division: \(6 \cdot 8 \div 4\)

Step 2: Perform Multiplication and Division from Left to Right

According to BODMAS/PEMDAS, division and multiplication must be performed from left to right. So first, multiply \(6 \cdot 8\) to get \(48\). The revised expression is: \(48 \div 4\).

Step 3: Perform the Division

Next, perform the division \(48 \div 4\) to get \(12\).

Key Concepts

Understanding BODMAS

The order of operations in mathematics is critical for solving expressions correctly. BODMAS is an acronym that helps students remember the sequence in which to perform operations within math problems. It stands for:

• Brackets
• Orders (i.e., powers and roots, such as square roots or exponents)
• Division
• Multiplication
• Addition
• Subtraction

This order means you should address any operations within brackets first, followed by orders (like exponents). After these, division and multiplication should be tackled, moving from left to right in the expression. Addition and subtraction are the last operations to solve, also handled from left to right. Understanding BODMAS ensures clarity when evaluating expressions and helps in avoiding errors. Remember, while BODMAS gives a guideline, division and multiplication operations are performed from left to right as they appear in the expression.

The Role of PEMDAS

Similar to BODMAS, PEMDAS is another acronym frequently used, particularly in countries like the United States, to remember the order of operations. The letters in PEMDAS stand for:

• Parentheses
• Exponents (e.g., powers and square roots)
• Multiplication
• Division
• Addition
• Subtraction

PEMDAS is essentially the same as BODMAS, with a slight variation in terminology. The vital point to remember here is that multiplication and division are of equal importance and should be performed from left to right, based on their position in the expression. The same rule applies to addition and subtraction. Using PEMDAS allows for consistent results when solving mathematical expressions and helps maintain accuracy. This method provides a comprehensive structure so that everyone solves expressions properly and uniformly.

Multiplication and Division

Within the framework of BODMAS and PEMDAS, multiplication and division are performed after any brackets or exponents have been solved. These operations are of equal precedence, which means you should handle them as they appear from left to right. In the expression \[6 \cdot 8 \div 4\], it's important to perform these operations sequentially, starting with the multiplication of 6 by 8 to get 48. Then, you carry out the division of 48 by 4 to reach the final result of 12. Remember:

• Do not jump to addition or subtraction before completing all multiplication and division tasks.
• Always go from left to right for both operations as part of the sequence.

Keeping this left-to-right rule in mind ensures you solve expressions correctly and efficiently, following the established order of operations.