Problem 85 - Intermediate Algebra

Question

Simplify each numerical expression. $$ 14.1-(17.2-13.6) $$

Step-by-Step Solution

Answer

The simplified expression is 10.5.

Step 1: Recognize and Address the Inner Parentheses

The expression includes parentheses, indicating that the operations inside them should be performed first. Our expression is \( 14.1 - (17.2 - 13.6) \). Begin by simplifying the expression within the parentheses: \( 17.2 - 13.6 \).

Step 2: Perform the Subtraction Inside the Parentheses

Calculate the difference in the inner expression: \( 17.2 - 13.6 = 3.6 \). Replace the result into the original expression: \( 14.1 - 3.6 \).

Step 3: Solve the Remaining Expression

Now, simplify the remaining expression: \( 14.1 - 3.6 \). Perform the subtraction to get: \( 14.1 - 3.6 = 10.5 \).

Key Concepts

Numerical Expression

A numerical expression in algebra is a combination of numbers and operations such as addition, subtraction, multiplication, or division. These expressions do not contain any variables, only specific numbers. In our exercise, the numerical expression is \( 14.1 - (17.2 - 13.6) \), which includes both numbers and the operation of subtraction.

• The purpose of a numerical expression is to calculate a numerical result.
• It's essential to understand that each number in the expression can represent actual values or quantities in real-life scenarios.
• The order in which you perform these operations is significant to solving the expression correctly.

It’s important to note that simplifying a numerical expression can involve several operations, but the process remains consistent: compute each operation step by step, following the rules of mathematics. This ensures you arrive at the correct solution.

Order of Operations

The order of operations is a set of rules that defines the sequence in which mathematical operations should be performed to ensure accurate results. This sequence is often remembered using the acronym PEMDAS:

• P: Parentheses - perform operations inside parentheses first.
• E: Exponents - solve exponentiation next.
• M and D: Multiplication and Division - from left to right.
• A and S: Addition and Subtraction - from left to right.

In our example, \( 14.1 - (17.2 - 13.6) \), we first address the parentheses as per the order of operations. This means simplifying \( 17.2 - 13.6 \) before anything else. By following these rules, you ensure calculations are both systematic and reliable.

Parentheses in Expressions

Parentheses in expressions play a crucial role in determining which operations need to be carried out first. They can change the outcome of a problem significantly by altering the original order of operations. In our example expression \( 14.1 - (17.2 - 13.6) \), the parentheses around \( 17.2 - 13.6 \) indicate that this subtraction must be performed before subtracting it from 14.1.

Using parentheses effectively can help:

• Clarify complex expressions and avoid ambiguity.
• Ensure certain parts of an expression are calculated first.
• Specify the intended order when multiple operations are involved.

This understanding allows you to break down complex problems into manageable parts, ensuring accuracy in final results. Always checking for parentheses first prevents errors and guarantees that every calculation step builds upon a correct foundation.