Problem 49 - Fundamentals of Mathematics

Question

Determine each value. $$ |-2|^{3} $$

Step-by-Step Solution

Answer

The value is 8.

Step 1: Understanding Absolute Values

The absolute value of a number is its distance from zero on the number line, regardless of direction. This means that any negative number becomes positive. Here, we need to find the absolute value of -2, which is 2.

Step 2: Calculating the Cube

After finding the absolute value, we need to raise it to the power of 3. So, we take the number 2 and calculate \(2^3\). This is done by multiplying 2 by itself three times: \(2 \times 2 \times 2 = 8\).

Key Concepts

Properties of Exponents

Exponents are an essential part of mathematics, and they help us understand how numbers behave when multiplied by themselves. An exponent tells us how many times we need to multiply the base number by itself. For instance, in the expression \(a^n\), \(a\) is the base and \(n\) is the exponent. This means \(a\) is multiplied by itself \(n\) times.

When dealing with exponents, it's crucial to know the key properties:

• **Product of Powers:** When multiplying two powers with the same base, you add their exponents. For example, \(a^m \times a^n = a^{m+n}\).
• **Power of a Power:** When raising a power to another power, you multiply the exponents: \((a^m)^n = a^{m\cdot n}\).
• **Power of a Product:** When you raise a product to a power, you raise each factor in the product: \((ab)^n = a^n \times b^n\).
• **Zero Exponent:** Any non-zero number raised to the power of zero equals 1: \(a^0 = 1\), provided \(a eq 0\).

These properties are particularly useful when simplifying expressions and solving equations involving exponents. They help us manage calculations efficiently and avoid mistakes.

Number Line

A number line is a visual representation of numbers laid out in a straight line. It extends infinitely in both directions, usually depicted horizontally. At the center of the number line, we have the point zero, which acts as a crucial reference:

• **Positive numbers** are placed to the right of zero.
• **Negative numbers** fall to the left of zero.
• The distance from any point on the line to zero indicates its absolute value.

The number line is an excellent tool for understanding concepts such as absolute values, addition, and subtraction of numbers. By visualizing how numbers are positioned, it becomes easier to grasp notions like order and relative size. When we talk about the absolute value, it represents how far a number is from zero, without considering direction. For instance, both -2 and 2 have an absolute value of 2, as they are two units away from zero.

Positive Numbers

Positive numbers play a fundamental role in mathematics and are part of the group of real numbers. They are placed to the right of zero on a number line and are always greater than zero. Here are some important points about positive numbers:

• **Natural Numbers:** These are the counting numbers like 1, 2, 3, etc., and they are inherently positive.
• **Non-Negative Numbers:** This category includes all positive numbers and zero, often used in contexts where negative values don't apply, like counting objects.

Positive numbers can be involved in various mathematical operations, such as addition, subtraction, multiplication, and division. When multiplied together, positive numbers produce a positive result. Additionally, when added or subtracted, their arrangement on the number line will illustrate the resulting value’s magnitude and direction. Understanding positive numbers is crucial for interpreting data, solving equations, and applying mathematical principles in real-world scenarios.