Problem 52 - Beginning and Intermediate Algebra

Question

Simplify each expression. (Remember the order of operations.) See Examples 4 and 5. $$ 4-6(7-3) $$

Step-by-Step Solution

Answer

-20

Step 1: Solve Inside the Parentheses

First, solve the expression inside the parentheses as part of the order of operations (PEMDAS/BODMAS). For the expression \((7-3)\), subtract 3 from 7 which equals 4. So, the expression becomes \(4 - 6 \times 4\).

Step 2: Multiply

Next, perform the multiplication. Multiply 6 by 4, which equals 24. Now the expression is simplified to \(4 - 24\).

Step 3: Perform Subtraction

Lastly, subtract 24 from 4 to find the final result. \(4 - 24 = -20\).

Key Concepts

Parentheses

Understanding parentheses is essential when dealing with mathematical expressions. Parentheses indicate which part of an expression should be calculated first. This concept is part of the order of operations, commonly remembered by the acronym PEMDAS or BODMAS, where 'P' or 'B' stand for parentheses or brackets respectively.

In the given exercise, the expression inside the parentheses is \((7 - 3)\). Here, subtraction is performed first because it is enclosed. This simplifies to 4. Remembering the role of parentheses can help avoid mistakes by ensuring that calculations are performed in the correct order. Incorrect handling of parentheses can lead to wrong outcomes.

• Always solve the operation inside the parentheses first.
• Once simplified, they can be dropped, and you focus on the remaining expression.

Multiplication

After handling the parentheses, the next step in our order of operations is multiplication. The importance of this step is highlighted in our exercise where the expression becomes \(4 - 6 \times 4\).

Here, we take the result from inside the parentheses, which is 4, and multiply it by 6. This results in 24. This step follows directly after evaluating parentheses or brackets, further underlining how crucial it is to follow the order of operations meticulously.

• Perform multiplication after simplifying expressions within parentheses.
• Multiply the number outside the parentheses by the result inside after simplification.

Subtraction

In the final part of this exercise, subtraction completes the simplification of the expression. The problem illustrates subtracting 24 from 4, resulting in \(4 - 24 = -20\).

Subtraction is often the last operation carried out after parentheses and multiplication have been handled. Knowing when to subtract is crucial, as performing this operation too early can lead to incorrect answers.

• Ensure all other operations such as parentheses and multiplication are complete before subtracting.
• Remember that subtraction can yield negative results, as seen in this exercise.

By following these guidelines, students can solve expressions consistently and accurately.