Problem 41 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ 1-6-7+8 $$

Step-by-Step Solution

Answer

The value of the expression is -4.

Step 1: Simplify the Expression

Start by simplifying the expression step by step without a calculator. Begin with the leftmost numbers, from left to right.First, calculate the result of the first two numbers:\[1 - 6 = -5\]

Step 2: Continue Simplifying

Now, take the result from the first part and subtract the next number in the sequence:\[-5 - 7 = -12\]

Step 3: Final Addition

Add the last number in the sequence to the result of the previous operation:\[-12 + 8 = -4\]

Step 4: Verify with a Calculator

To ensure accuracy, use a calculator to verify the result by entering the full expression:\[1 - 6 - 7 + 8\]The calculator should confirm that the result is \[-4\].

Key Concepts

Expression Simplification

When faced with a complex arithmetic expression, the goal is to break it down into simpler parts. This process, known as expression simplification, involves gradually reducing the expression to a single value. In the example provided, the expression to simplify was \(1 - 6 - 7 + 8\). Simplification usually starts from the leftmost side of the expression, executing each operation in sequence. By performing each arithmetic operation one step at a time, the problem becomes more manageable. This method helps in reducing errors and understanding the logical flow of operations. As each part of the expression is simplified, you get closer to the final solution.

Order of Operations

The concept of order of operations is essential when dealing with mathematical expressions that contain multiple operations. This set of rules, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial for determining the correct order in which operations should be performed.

• Parentheses: Complete operations inside parentheses first.
• Exponents: Resolve any numbers with exponents.
• Multiplication and Division: Work from left to right, and perform all multiplication and division operations.
• Addition and Subtraction: Finally, perform addition and subtraction from left to right.

In the example \(1 - 6 - 7 + 8\), we only have addition and subtraction, which means we proceed from left to right since there are no parentheses or exponents. It's important to note that addition and subtraction have the same precedence, so they are solved as they appear from left to right.

Using a Calculator

After simplifying a mathematical expression manually, it’s always a good idea to verify your results using a calculator. This ensures accuracy and confirms that no errors were made during manual calculations. For our expression \(1 - 6 - 7 + 8\), enter it directly into a calculator as is. Most calculators follow the order of operations internally, so there's little risk of mistakes if you input the expression correctly.

When using a calculator, double-check to see that each number and operator is entered correctly. Make sure your calculator is not set to any specific modes that might alter the outcome, such as financial functions or alternate bases apart from the usual decimal base. By checking the result with a calculator, you can be confident in the accuracy of your solution. In this case, the calculator confirms the result to be \(-4\), reassuring us that our manual calculations were correct.