Problem 17 - Intermediate Algebra

Question

Simplify each numerical expression. Be sure to take advantage of the properties whenever they can be used to make the computations easier. $$ [83+(-99)]+18 $$

Step-by-Step Solution

Answer

The simplified result is 2.

Step 1: Understand the Expression

We need to simplify the expression \([83+(-99)]+18\). This involves performing arithmetic operations while keeping in mind properties like addition of integers.

Step 2: Apply the Associative Property

The associative property allows us to group numbers differently when adding. We have \([83 + (-99)] + 18\), which can be simplified by first calculating \(83 + (-99)\).

Step 3: Solve the Inner Expression

Calculate \(83 + (-99)\). This simplifies to \(83 - 99\), since adding a negative number is the same as subtraction. Thus, \(83 - 99 = -16\).

Step 4: Complete the Addition

Now, substitute \(-16\) back into the expression in place of \([83 + (-99)]\). We have \(-16 + 18\).

Step 5: Calculate Final Result

Performing the addition \(-16 + 18\) gives us \(2\).

Key Concepts

Associative Property

The associative property of addition states that how we group numbers in an addition problem does not change the sum. This means we can alter the grouping of numbers to make calculations easier, as long as we are only dealing with addition. For example, if we have the expression \( (a + b) + c \), the associative property tells us that this is equivalent to \( a + (b + c) \). The same logic applies in reverse.

• Helpful when needed to simplify complex expressions.
• Allows focusing on simplifying easier parts first.

In our original exercise \( [83+(-99)]+18 \), using the associative property lets us group \( 83 + (-99) \) first, simplify it, and then add \( 18 \) to the result. This step-by-step approach leverages the associative property to make the entire calculation more manageable.

Arithmetic Operations

Arithmetic operations are basic calculations including addition, subtraction, multiplication, and division. In this problem, we're primarily focused on addition and subtraction because we are simplifying a numerical expression with integers. Understanding these operations is crucial as they form the foundation for more complex math problems.For addition, numbers are combined to form a sum. Conversely, subtraction involves finding the difference between numbers. Adding a negative number can be thought of as subtraction, which is exactly what we did in the step \( 83 + (-99) \). By thinking of it as \( 83 - 99 \), it becomes a straightforward subtraction problem. As you practice, these arithmetic operations become automatic, but remember:

• Ensure you understand whether you're adding or subtracting.
• Keep track of positive and negative signs.
• Apply operations step-by-step for clarity.

Addition of Integers

Addition of integers involves combining whole numbers, which can include both positive and negative values. When adding integers, it's important to pay attention to their signs.With integers:

• If the numbers have the same sign, add their absolute values and keep the sign.
• If the numbers have different signs, subtract the smaller absolute value from the larger one and keep the sign of the larger absolute value.

In the exercise \( [83+(-99)]+18 \), \( 83 + (-99) \) involves addition of integers with different signs. You compare the absolute values:

- Absolute value of 83 is 83
- Absolute value of -99 is 99Since 99 is larger, the result of the operation will take the negative sign, resulting in \(-16\). Finally, adding 18 to this result,we use the same rules.

This careful attention to adding integers helps avoid errors and ensures you're on the right path to solving even more complex equations in the future.