Problem 31 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the order of operations to simplify each expression. $$45 \div 5 \cdot 3$$

Step-by-Step Solution

Answer

The simplified expression is 27.

Step 1: Identify Operations

The given expression: \(45 \div 5 \cdot 3\) includes both division and multiplication.

Step 2: Apply Division

According to the order of operations, need to perform the division before multiplication. By dividing 45 by 5, the expression simplifies to \(9 \cdot 3\).

Step 3: Apply Multiplication

Now, multiply 9 by 3 to get 27. The simplified expression is now 27.

Key Concepts

Division

The division is crucial when simplifying expressions, especially when multiple operations are present. In any given expression, division often pairs with other operations like multiplication. When you see division in an expression, it is vital to handle it correctly following the order of operations, which guides us in solving mathematical problems. According to this order, simplifications should prioritize division and multiplication before tackling addition or subtraction, from left to right. This sequence ensures accuracy and consistency.

• Always locate the parts of the expression being divided.
• Perform the division first if it appears before multiplication, as in the expression \(45 \div 5 \cdot 3\).

Only after addressing division do we move on to any multiplication present.

Multiplication

Multiplication typically comes after division when simplifying expressions, following the order of operations. In expressions mingling both multiplication and division, like \(45 \div 5 \cdot 3\), we apply multiplication following the completion of any division. This approach ensures each step is approached logically and methodically.

• Check if multiplication is next after any initial division.
• Complete the multiplication: take the result from the division step, and proceed with multiplying \(9 \cdot 3\).

Understanding this flow of operations helps you achieve the correct answer efficiently.

Simplify Expression

Simplifying an expression means breaking it down to its simplest form by performing all necessary operations such as division and multiplication. The key is the order of operations, often remembered through the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). For the expression \(45 \div 5 \cdot 3\), the absence of parentheses or exponents directs us straight to division and multiplication.

• Perform operations strictly in the defined order: Division first, then multiplication.
• Step-by-step simplification: First, compute \(45 \div 5\), then use the result to conduct \(9 \cdot 3\).

The expression is ultimately reduced to a single number, demonstrating the power of proper sequence in simplification.