Problem 56 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$21-4 \cdot 3+2$$

Step-by-Step Solution

Answer

The simplified expression is 11.

Step 1: Follow the Order of Operations

First, remember that mathematical expressions must be simplified by following the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Step 2: Perform Multiplication

Identify any multiplication in the expression. Here, we have the multiplication \(4 \cdot 3\). Perform the multiplication: \(4 \cdot 3 = 12\).

Step 3: Replace the Multiplication Result

Substitute the result of the multiplication back into the expression. The expression now becomes \(21 - 12 + 2\).

Step 4: Perform Subtraction

Next, perform the subtraction operation from left to right. Calculate \(21 - 12 = 9\).

Step 5: Perform Addition

Finally, add the remaining numbers: \(9 + 2 = 11\).

Step 6: Final Simplified Expression

The expression has been simplified to \(11\).

Key Concepts

PEMDAS

In order to confidently tackle mathematical expressions, it is crucial to understand the order of operations. The acronym **PEMDAS** helps us remember the specific sequence we must follow:

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

This sequence is essential to ensure we simplify expressions correctly. Always work through expressions step by step. Begin by calculating inside any parentheses, then move on to exponents. Next, perform multiplication and division from left to right, before finally tackling addition and subtraction, also from left to right. This method allows for consistent results, regardless of the complexity of the initial expression.

Multiplication

Multiplication is one of the core operations in math that allows us to quickly find the result of adding a number to itself multiple times. In our expression, we identify multiplication as the first operation to perform after any parentheses or exponents, due to PEMDAS.Consider the multiplication part of the expression: \(4 \cdot 3\). Here, we multiply 4 by 3. This simplifies to 12 because adding four 3's together (3 + 3 + 3 + 3) gives us the same product. Once multiplication is complete, replace the result back into the expression, simplifying it. Understanding and identifying multiplication at the right step is fundamental for accurate calculations.

Addition and Subtraction

Addition and subtraction, though basic, are essential operations in simplifying equations. According to PEMDAS, these operations are the final steps in our calculation order, and they should be performed from left to right.In our exercise, once multiplication is completed, we're left with \(21 - 12 + 2\). First, perform the subtraction: 21 subtract 12 gives us 9. Then, we add the last number, which is 2, to 9.Sequentially tackling addition and subtraction ensures accuracy. Remember, these two operations are like the finishing touches in simplifying expressions, wrapping up all previous calculations into the final, simplest form.

Simplifying Expressions

Simplifying expressions make complex problems much easier to handle by breaking them down into basic arithmetic operations. The aim is to condense these operations into a single, straightforward number or expression.The expression in this exercise, \(21 - 4 \cdot 3 + 2\), undergoes several steps of simplification. Starting with multiplication to obtain 12, we then follow up with subtraction to reach 9, and finalize with addition to get 11.By simplifying, we transform a complicated initial expression into a comprehensible answer, in this case, 11. Simplicity not only helps with understanding but also aids in checking work for accuracy. Always ensure that each step adheres to the order of operations for the most reliable result.