Problem 37 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the expression. $$ |\sqrt{2}-1|+|3-\sqrt{2}| $$

Step-by-Step Solution

Answer

The given expression is \( | \sqrt{2} - 1 | + | 3 - \sqrt{2} |\). Evaluating the expressions inside the absolute value brackets and finding the absolute values, we have: 1. \(| \sqrt{2} - 1 | ≈ | 1.41 - 1 |= 0.41\) 2. \(| 3 - \sqrt{2} | ≈ | 3 - 1.41 |= 1.59\) Adding the absolute values, we get: \(0.41 + 1.59 = 2\). Therefore, the expression evaluates to 2.

Step 1: Understand the expression

We want to evaluate the following expression: \( | \sqrt{2} - 1 | + | 3 - \sqrt{2} |\). To do this, we will first find the values of each expression inside the absolute value brackets and then take the absolute values. After that, we will sum the individual absolute values to find the final result. Step 2: Evaluate the expressions inside the absolute value brackets

Step 2: Evaluate the expressions

To calculate the expressions inside the absolute value brackets, we have: 1. \(\sqrt{2} - 1\) 2. \(3 - \sqrt{2}\) Step 3: Find the absolute values

Step 3: Compute the absolute values

Next, we will find the absolute values of our calculated expressions: 1. \(| \sqrt{2} - 1 | ≈ | 1.41 - 1 |= | 0.41 | = 0.41\) 2. \(| 3 - \sqrt{2} | ≈ | 3 - 1.41 |= | 1.59 | = 1.59\) Step 4: Add the absolute values

Step 4: Sum the absolute values

Finally, we will add the absolute values together: \(0.41 + 1.59 = 2\) So, the expression evaluates to 2.

Key Concepts

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. It is one of the fundamental operations in mathematics. Typically, the square root is denoted by the radical symbol "\(\sqrt{}\)". For example, the square root of 4 is 2, because \(2 \times 2 = 4\). Similarly, the square root of 9 is 3, because \(3 \times 3 = 9\).

It's important to note that not all numbers have a simple integer as their square root. In this exercise, we deal with \(\sqrt{2}\), which is an irrational number. Its approximate value is 1.41, and it cannot be expressed as a simple fraction.

• The square root helps in expressing a number in terms of its power, specifically, the power of 1/2.
• Understanding square roots is crucial for solving many mathematical problems, especially when dealing with powers and roots.

Knowing how to approximate a square root is an essential skill, especially when you want to simplify or evaluate expressions like \(|\sqrt{2} - 1|\).

Expression Evaluation

In mathematics, expression evaluation is the process of, well, figuring out what a given expression equals! This exercise specifically asks us to evaluate an expression combining square roots and absolute values.

To evaluate the expression \(|\sqrt{2} - 1| + |3 - \sqrt{2}|\), you follow these basic steps:

• First, deal with the operation inside each absolute value by substituting and calculating the numbers involved.
• Then, transform these results using the absolute value, removing any negative signs.
• Finally, sum up the simplified results to solve the entire expression.

Evaluating expressions systematically like this ensures you achieve a correct result. It emphasizes the importance of order in performing mathematical calculations, ensuring every step logically follows the previous one.

Mathematical Operations

Mathematical operations are the building blocks of working with numbers and expressions. They include functions like addition, subtraction, multiplication, division, and many more. This exercise prominently features the use of absolute values and square roots as core operations.

• Absolute value, denoted by vertical bars \(|\cdot|\), provides the distance of a number from zero, removing any negative sign.
• Square roots simplify or transform numbers when evaluating powers.

In this problem, you saw how absolute values were used in combination with subtraction to first evaluate \(\sqrt{2} - 1\) and \(3 - \sqrt{2}\), transforming these results to positive values, which simplifies and provides the total of the combined expression.

Understanding these operations makes it easier to tackle more complex math problems, giving you comprehensive tools to solve numerous equations.