Problem 32 - Fundamentals of Mathematics

Question

Determine each of the values. $$ |-2|^{3} $$

Step-by-Step Solution

Answer

8

Step 1: Solve Absolute Value

The expression involves an absolute value. The absolute value of a number \(x\) is the distance between \(x\) and 0 on the number line, without considering direction. Therefore, \(|-2| = 2\), because the distance is 2.

Step 2: Perform Exponentiation

Now that we have \(|-2| = 2\), the expression becomes \(2^3\). To calculate the value, we multiply the base 2 by itself 3 times: \(2 \times 2 \times 2\). This equals 8.

Key Concepts

Exponentiation

Exponentiation is a powerful mathematical operation that involves raising a number, called the base, to a certain power, known as the exponent. The process is a form of repeated multiplication. For example, when we have an expression like \(2^3\), the base is 2 and the exponent is 3. This means we multiply the base by itself a number of times equal to the exponent.

The steps to calculate \(2^3\) are as follows:

• Multiply 2 by itself: \(2 \times 2 = 4\)
• Take the result and multiply it by the base once more: \(4 \times 2 = 8\)
• Thus, \(2^3 = 8\)

Exponentiation allows us to concisely express large numbers and solve problems involving growth or decay. It's a key concept in algebra and is essential for understanding more complex mathematical ideas.

Number Line

The number line is a straightforward, visual representation of numbers laid out in a straight line. It is an essential tool in mathematics for understanding number sequences and operations. Each point on the line corresponds to a numerical value, from negatives to positives.

Here's how you can navigate and understand a number line:

• The line extends infinitely in both directions.
• Zero is typically placed in the center. To the right are positive numbers, and to the left are negative numbers.
• Each equally spaced point on the line indicates consecutive integers or fractions.

Using a number line helps visualize addition and subtraction as movements to the right or left. It's also particularly helpful for understanding the concepts of magnitude and direction, especially when assessing the distance of numbers from one another.

Distance on the Number Line

Distance on the number line represents how far apart two points or numbers are, regardless of direction. The idea is straightforward: a measure of separation which can be easily visualized.

Calculating this distance involves the following steps:

• Identify the two numbers between which you want to find the distance.
• Calculate the absolute difference between these numbers (ignore negative signs). For example, the distance between -3 and 2 is \(|-3-2| = |5| = 5\).
• The absence of direction in measuring distance is the hallmark of absolute value.

Understanding distance on the number line is crucial for solving mathematical problems involving intervals and spans across positive and negative numbers. It further extends to understanding the absolute value, which is essentially just the distance of a number from zero.