Problem 93 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ 25-[10-(3-12)] $$

Step-by-Step Solution

Answer

6

Step 1: Simplify inside the innermost parentheses

Begin by simplifying the expression inside the innermost parentheses: \(3 - 12\). This simplifies to \(3 - 12 = -9\). The expression now becomes \(25 - [10 - (-9)]\).

Step 2: Simplify the expression within the brackets

Next, simplify the expression inside the brackets: \[10 - (-9)\]. Subtracting a negative number is equivalent to adding the positive: \[10 - (-9) = 10 + 9 = 19\]. The expression now is simplified to \(25 - 19\).

Step 3: Subtract the simplified value from 25

Finally, perform the subtraction: \(25 - 19\). This calculates to \(25 - 19 = 6\).

Key Concepts

Order of Operations

When simplifying mathematical expressions, it's crucial to follow the order of operations, often remembered by the acronym PEMDAS:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

This order dictates which operations to perform first to accurately solve the expression. Failing to follow PEMDAS can lead to incorrect results.
For instance, in the exercise \(25 - [10 - (3 - 12)]\), we follow the order of operations step-by-step:
1. Simplify inside parentheses
2. Perform operations within brackets
3. Complete any addition or subtraction last
By adhering to PEMDAS, we get the correct result of 6.

Parentheses

In mathematical expressions, parentheses are used to indicate which operations should be performed first. Always simplify the innermost parentheses before moving outward.
Consider the expression \(25 - [10 - (3 - 12)]\):

• Begin with the innermost part, \((3 - 12)\), which simplifies to \(-9\).
• Next, address the brackets \([10 - (-9)]\).

Using parentheses helps to clarify priority in calculations and ensures that complex expressions are broken down step-by-step. This technique avoids mistakes and provides a clear path to the solution. For more complex expressions, nesting multiple parentheses within each other can occur but always resolve them from the innermost part outward.

Negative Numbers

Dealing with negative numbers can be tricky. Here are some key points:

• A negative number is a number less than zero, such as -5.
• Subtracting a negative number is the same as adding its positive counterpart.

For example, in our exercise \(25 - [10 - (-9)]\), you encounter subtracting a negative number:\br>

• \(10 - (-9)\) becomes \(10 + 9\).

This step simplifies the expression and makes it easier to handle. Negative numbers can change the direction of the operation and must be handled carefully. Practice with different examples helps in mastering the concept and makes solving expressions involving negatives much smoother.