Problem 48 - Fundamentals of Mathematics

Question

Determine each value. $$ |-5|^{2} $$

Step-by-Step Solution

Answer

The value is 25.

Step 1: Understand the Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering direction. For example, the absolute value of \(-5\) is \(5\) because it's 5 units away from zero.

Step 2: Apply the Absolute Value

Substitute the absolute value of \(-5\) into the expression to get \(5^{2}\).

Step 3: Square the Result

Calculate \(5^{2}\) by multiplying 5 by itself. This results in \(5 imes 5 = 25\).

Key Concepts

Distance from zero

The concept of absolute value revolves around the idea of 'distance from zero.' Imagine you are looking at a number line. Numbers are placed on this line based on their magnitude and direction from zero, the middle point. The absolute value is the numeric distance from zero, disregarding any negative signs.
This means if you have the number -5, its absolute value is simply 5 because it's five units away from zero. This principle is crucial because it helps us understand that absolute value measures the size of a number without considering whether it points to the positive or negative direction on the number line.

Number line

A number line is a visual representation that helps us understand numbers and their positions relative to each other. Zero is at the center, with positive numbers extending to the right and negative numbers to the left. This visualization is key in grasping the concept of absolute value.
When evaluating the absolute value, what we're essentially doing is measuring the distance from zero along this line. For -5, we find its position to the left of zero, count 5 units, and understand that its absolute value is 5. This graphical tool stimulates comprehension by allowing us to picture numbers in relation to a fixed point.

Squared value

Squaring a number is a mathematical operation where a number is multiplied by itself. It is represented often by an exponent of 2. For example, when we evaluate \|-5\|^{2}, we first focus on the absolute value, which reassures us that we're working with a positive value, i.e., 5.
Then, squaring this number involves calculating \(5^2\), which is 5 multiplied by itself. So when we do \(5 \times 5\), we get 25. The notion of squaring amplifies a number's magnitude and is integral to various mathematical contexts, offering a path to exponential growth in equations and functions.