Problem 63 - Intermediate Algebra

Question

Simplify each numerical expression. $$ [14-(16-18)]-[32-(8-9)] $$

Step-by-Step Solution

Answer

The simplified expression is \(-17\).

Step 1: Simplify the Inner Parentheses

First, we'll simplify the expressions inside the inner parentheses. Begin with \( (16-18) \). The result of \( 16-18 \) is \(-2\).

Step 2: Simplify the Inner Parentheses

Next, simplify the other inner parentheses \( (8-9) \). The result of \( 8-9 \) is \(-1\).

Step 3: Replace and Simplify Outer Parentheses

Now, replace the simplified results in the original expression: \[ [14-(-2)] - [32-(-1)] \]. Simplify the expressions inside the brackets:

Step 4: Simplify Brackets

The expression \( 14 - (-2) \) becomes \( 14 + 2 = 16 \). For the second bracket: \( 32 - (-1) \) becomes \( 32 + 1 = 33 \).

Step 5: Solve the Remaining Expression

Now replace the simplified brackets into the expression: \( 16 - 33 \) Subtract to find: \( 16 - 33 = -17 \).

Key Concepts

Numerical Expression

When we talk about a numerical expression, it's essentially a combination of numbers, operations (like addition or subtraction), and sometimes parentheses. The main goal when dealing with a numerical expression is to simplify it, which means combining the elements by performing the necessary operations. For example, with the expression \[ 14 - (16 - 18) \] compared to something more straightforward like \( 3 + 5 \), the complexity increases due to the brackets involved.
What differentiates these expressions is the need for careful calculation by following specific rules, such as the order of operations, to ensure you arrive at the correct number.

Order of Operations

The order of operations is vital to solving numerical expressions correctly. It's like a manual that tells you which math operation to perform first, second, and so on. The order is commonly remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the given expression \[ [14 - (16 - 18)] - [32 - (8 - 9)] \], we first resolve operations inside the parentheses according to PEMDAS. This strategy ensures calculations are performed in the right sequence, leading ultimately to a single, simplified number.

Parentheses

Parentheses play a critical role in determining the order of operations within a numerical expression. They signal that the operations contained within them should be performed first. When simplifying an expression, make sure to:

• Identify the innermost parentheses first.
• Perform the operations within these parentheses.

In the example \( (16 - 18) \), we address this calculation initially because of the parentheses. Solving it yields \(-2\).
This crucial first step sets the stage for correctly managing the more extensive expression, as miscalculating here can affect the entire solution.

Subtraction

Subtraction is a fundamental operation that involves taking away one number from another. It's crucial to execute it correctly, especially when dealing with negative numbers or nested operations. In the expression \[ [14 - (-2)] - [32 - (-1)] \], all subtractions need careful attention to the signs:

• Removing a negative number is equivalent to adding the positive counterpart.

Thus, \( 14 - (-2) \) translates into \( 14 + 2 = 16 \). Similarly, \( 32 - (-1) \) becomes \( 32 + 1 = 33 \).
Finally, to find the result, you perform the final subtraction: \( 16 - 33 = -17 \). Each step requires attention to detail, ensuring no sign changes are overlooked.