Problem 57 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ -(8-21) $$

Step-by-Step Solution

Answer

The value is 13.

Step 1: Identify the Expression

The given expression to evaluate is \[-(8-21).\] Identify that there is a negative sign in front of the parentheses containing the subtraction 8 - 21.

Step 2: Perform the Subtraction Inside the Parentheses

Calculate the subtraction inside the parentheses: \[8 - 21 = -13.\]Now, the expression simplifies to \(-(-13)\).

Step 3: Apply the Negative Sign Outside

Since there is a negative sign in front of \(-13\), we apply it:\[-(-13) = 13.\]This changes the sign of \(-13\) to \(+13\).

Step 4: Verify Using a Calculator

Enter \[-(8-21)\] into a calculator. The calculator should display \(13\). This verifies our manual calculation is correct.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers with a minus sign in front of them. They are less than zero, which means they represent values below zero. For example, in temperature,
negative numbers might represent temperatures below freezing. Negative numbers can be tricky because they reverse the usual order of numbers.
For instance:

• -2 is less than 0
• -5 is less than -2

When dealing with equations that involve negative numbers, it's important to carefully follow any addition or subtraction rules. Watching out for these rules helps avoid mistakes, especially in combined operations like those in the given expression. Remember, subtracting a negative number is like adding a positive number, as signs change.

Simplifying Subtraction

Subtraction may seem simple, but it's a core part of many algebraic expressions.
In the expression \(-(8-21)\), focus on the subtraction \(8 - 21\) first.
Here, the trick is to recognize that 21 is greater than 8, so the result of this subtraction would be a negative number.

• 8 minus 21 can be thought of as moving 21 steps backward from 8 on a number line, ending at -13

This idea becomes especially useful when calculating differences where the second number is larger. Always pay close attention to the order of numbers; it significantly impacts the result, often resulting in negative outcomes.

Parentheses and Their Role in Order of Operations

Parentheses play a crucial role in mathematical expressions as they dictate what calculations are priority. The basic rule in math is to perform operations inside parentheses first, before handling other operations like addition or subtraction outside them.
When you encounter an expression like \(-(8-21)\), compute the subtraction inside the parentheses first to simplify the expression.
Only after this step do you handle any operations outside of the parentheses. The parentheses essentially create a mini-problem to solve within the broader problem. This segmentation makes complex problems more manageable by breaking them down into smaller, easier-to-tackle components.

Using a Calculator Effectively

Calculators are valuable tools for verifying manual calculations and handling complex computations. They ensure accuracy and can save time, especially when checking results.
For expressions such as \(-(8-21)\), simply input the operation as shown into your calculator. Include parentheses explicitly to ensure that all arithmetic within them is done first. Here's how to do it:

• Press the open parenthesis button
• Input the numbers and operation: `8 - 21`
• Close the parenthesis
• Apply the negative sign in front

Using the calculator in this manner helps confirm your manual calculations, providing peace of mind by double-checking the accuracy of complicated or susceptible-to-error operations.