Problem 21 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(4 \cdot 8-6 \cdot 2\)

Step-by-Step Solution

Answer

The simplified expression is 20.

Step 1: Identify the Operations

The expression includes two operations - multiplication and subtraction. The given expression is \(4 \cdot 8 - 6 \cdot 2\).

Step 2: Perform the Multiplications

First, perform the multiplications in the expression. Calculate \(4 \cdot 8 = 32\) and \(6 \cdot 2 = 12\).

Step 3: Substitute the Results

Replace the multiplication results back into the original expression. The expression now becomes \(32 - 12\).

Step 4: Perform the Subtraction

Subtract 12 from 32. Calculate \(32 - 12 = 20\).

Step 5: Conclusion

The simplified expression is \(20\).

Key Concepts

Order of Operations

When simplifying expressions in math, especially those with multiple operations, following the correct order is crucial. This is known as the "order of operations." This concept helps students correctly solve expressions like the given example:

• The expression is: \(4 \cdot 8 - 6 \cdot 2\).
• In operations, multiplication and division take precedence, followed by addition and subtraction.
• This rule is often remembered using the acronym PEMDAS: Parentheses, Exponents, Multiplication/Division (from left to right), Addition/Subtraction (from left to right).

First, identify any operations: multiplication and subtraction in this case. According to PEMDAS, multiplication should be dealt with before subtraction. Hence, carry out all multiplication steps before addressing subtraction. This approach ensures accurate and consistent results.

Multiplication

Multiplication is one of the key operations you must address first in the expression. Let's break down how to handle these steps:

• In the expression \(4 \cdot 8 - 6 \cdot 2\), perform each multiplication separately.
• Calculate \(4 \cdot 8\), which yields \(32\).
• Next, compute the second multiplication \(6 \cdot 2\), resulting in \(12\).

Once you have these results, substitute them back into the expression, replacing the initial multiples. Now, you have simplified the expression to \(32 - 12\). Multiplication simplifies expressions by effectively scaling numbers, and it's essential to complete this step before moving on to other operations.

Subtraction

Subtraction is typically carried out after multiplication when simplifying expressions. Consider the following steps:

• After carrying out the multiplications, you are left with the expression: \(32 - 12\).
• Now, simply perform the subtraction: \(32 - 12\).
• This calculation gives you the final answer, \(20\).

Subtraction is crucial for reducing an expression to its simplest form. It's the final step in the process, ensuring that all elements have been effectively simplified. By systematically following each operation's rules, you can confidently simplify even the most complex mathematical expressions.