Problem 34 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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

Step-by-Step Solution

Answer

The value of the given expression is \(1 + 2\sqrt{2}\).

Step 1: Evaluate the absolute values

First, we need to evaluate the absolute values. The absolute value of a real number is its distance from 0 on the number line. It is always positive or zero. Therefore, we have \[ |-1| = 1 \] and \[ |-2|= 2 \]

Step 2: Calculate the square root

Now we have to calculate the square root of 2. Since it is a constant and not given as a variable, we will keep it as is: \[ \sqrt{2} \]

Step 3: Perform the addition

Now that we have evaluated the absolute values and identified the square root, we can perform the addition as indicated. So, replacing the values in the expression, we get: \[ 1 + \sqrt{2} \times 2 \] Now, multiply the square root with the constant: \[ 1 + 2\sqrt{2} \] Thus, the value of the given expression is \(1 + 2\sqrt{2}\).

Key Concepts

Absolute Value

Absolute value is a mathematical concept that denotes the distance of a number from zero on the number line.
It is symbolized by vertical bars around the number, like \(|a|\).
Here are a few key points about absolute value:

• Absolute value converts any number into its non-negative equivalent.
• For positive numbers, the absolute value is the number itself. For negative numbers, it is the positive counterpart.
• The absolute value of zero is zero.

Understanding absolute value is important because it does not simply remove the negative sign but considers the concept of distance.
In the context of the exercise, to find \(|-1|\) and \(|-2|\), you are finding how far \-1\ and \-2\ are from zero.
Their distances are both positive, resulting in \(1\) and \(2\) respectively. Absolute value plays a crucial role when dealing with measurements, physics, and real-world problems.

Square Root

The square root is a mathematical operation that asks, "What number, when multiplied by itself, gives the original number?"
It is symbolized by the radical sign (\(\sqrt{}\)).
Let’s break down some crucial points about square roots:

• The square root of a number \(x\) is another number \(y\) such that \(y^2 = x\).
• Square roots are essential in various applications, including geometry and quadratic equations.
• Certain numbers have exact square roots, such as \( \sqrt{4} = 2\).
• Other numbers, like \(\sqrt{2}\), are irrational and are approximated as 1.414... in decimal form.

In our exercise, we deal with \(\sqrt{2}\), a common irrational number.
It is typically left in its square root form unless an approximation is suitable for the problem at hand.
Understanding square roots is essential for solving a big variety of algebraic and geometric problems.

Mathematical Expressions

A mathematical expression is a combination of numbers, variables, and operators that represent a value.
Here are some fundamental points:

• Expressions can include operations like addition, subtraction, multiplication, and division.
• The expression must be simplified to find its value.
• Expressions can also include functions like absolute value and square roots.
• It’s crucial to perform operations in the correct order as per PEMDAS/BODMAS rules.

In the exercise, the expression \(|-1| + \sqrt{2} |-2|\) involves both absolute values and a square root.
By simplifying each part, first calculating absolute values, followed by the square root, and then performing addition and multiplication, we find the expression to resolve to \(1 + 2\sqrt{2}\).
Breaking down expressions is key in mathematics as it unveils the path to understanding relationships and structures within equations.