Problem 33 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the order of operations to simplify each expression. $$6 \cdot 8 \div 4$$

Step-by-Step Solution

Answer

The simplified expression is 12.

Step 1: Perform the multiplication

Here, you first perform the multiplication between 6 and 8, which results in 48.

Step 2: Perform the division

Then, divide the result of the multiplication (48) by 4. This operation produces 12.

Key Concepts

Order of Operations

When simplifying algebraic expressions, the order in which you perform calculations matters greatly. This sequence is essential to ensure that everyone arrives at the same answer when solving the same mathematical expression. Without a universal order, expressions could yield multiple answers depending on the individual approach.

To avoid such confusion, we follow the order of operations, a set of rules that defines the sequence to perform operations in an expression. This sequence goes by the acronym PEMDAS or BODMAS, which stand for: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and finally, Addition and Subtraction (from left to right). The key is to work from the innermost parentheses outwards and handle multiplication or division as they appear from left to right within the expression.

Algebraic Multiplication

In algebra, multiplication is a fundamental operation where two or more numbers, known as factors, are combined to get a product. When working with quantities that include variables, the principles of multiplication remain the same, but it's essential to keep track of coefficients (the numerical part) and the variables (the letter part).

For example, when multiplying two algebraic terms like 3x and 2y, the result would be 6xy, where you multiply the coefficients (3 and 2) and then write the variables next to each other. Remember, when there are no visible operators between a number and a variable, or between two variables, it implicitly means to multiply them.

Division in Algebra

Just as important as multiplication is division in algebra. It represents the distribution of a quantity into equal parts, and when it comes to algebraic expressions, the approach mimics that of numerical division. Always divide the coefficients (numbers) and simplify the variables while following proper algebraic laws.

Dividing by zero is undefined, and expressions should be simplified as much as possible to their lowest terms. For example, if you're dividing 16x by 4, you divide the coefficient (16) by the divisor (4) to get 4x. Take extra care with negative signs and variable exponents during division to avoid common mistakes.

PEMDAS/BODMAS Rules

The PEMDAS/BODMAS rules are a mnemonic that helps remember the order of operations in mathematical expressions. The acronyms stand for Parentheses/Brackets, Exponents/Orders, Multiplication, Division, Addition, and Subtraction, which represents the hierarchy of operations.

Examples Using PEMDAS/BODMAS:

• Parentheses/Brackets: Solve any operations within parentheses or brackets first.
• Exponents/Orders: Calculate powers or roots second.
• Multiplication/Division: Next, perform all multiplication and division as they appear from left to right.
• Addition/Subtraction: Finally, handle addition and subtraction from left to right.

It's crucial to move through this order accurately to simplify expressions correctly. Any deviation can lead to incorrect results.