Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-10,\) tell whether each represents a number that is rational, irrational, or neither. $$ \sqrt[3]{-8} $$

Step-by-Step Solution

Answer

The expression represents a rational number.

Step 1: Identify the Operation

The given expression is \( \sqrt[3]{-8} \), which is a cube root. A cube root asks what number, when multiplied by itself three times, results in the original number, in this case, \(-8\).

Step 2: Calculate the Cube Root

To find \( \sqrt[3]{-8} \), we need to find a number \( x \) such that \( x^3 = -8 \). Since \(-2^3 = -2 \times -2 \times -2 = -8\), we have \( \sqrt[3]{-8} = -2 \).

Step 3: Assess the Type of Number

The result \(-2\) is an integer. All integers are also rational numbers because they can be expressed as a fraction with a denominator of 1, e.g., \(-2 = \frac{-2}{1}\). Therefore, \( \sqrt[3]{-8} \) is rational.

Key Concepts

Cube Roots

A cube root helps us figure out which number, when multiplied by itself three times, gives us a specific value. Think of this operation as the opposite of cubing a number. For instance, if you are given \( \sqrt[3]{-8} \), the task is to find a number \( x \) satisfying \( x^3 = -8 \).

In this case, \(-2\) is the answer because \(-2 \times -2 \times -2 = -8\). Cube roots can be calculated for both positive and negative numbers, making them distinct in this way, as square roots do not have real solutions for negative under the real number system.

By understanding cube roots, you can solve many real-world and mathematical problems related to volumes and geometry, such as finding the side length of a cube when you know its volume.

Integer

Integers are an essential part of mathematics. Simply put, integers are whole numbers that can be either positive, negative, or zero. What makes them unique is that they do not include any fractional or decimal parts. For example, the numbers \(-3, 0,\) and \(7\) are all integers.

The solution to \( \sqrt[3]{-8} \) was \(-2\), which is an example of an integer.

Integers are used every day in real life, like when you count items, temperatures, or money. Understanding integers is crucial because they form the building blocks for more complex number types and concepts in math.

Rational Number Identification

A rational number can be defined as any number that can be expressed as the quotient or fraction \( \frac{p}{q} \) of two integers, where \( p \) and \( q \) are integers and \( q eq 0 \). This means rational numbers include fractions like \( \frac{1}{2} \), whole numbers like \( 4 \), and repeating decimals like \( 0.333... \).

In the given example, \( \sqrt[3]{-8} = -2 \), which is a rational number because it can be written as \( \frac{-2}{1} \).

Understanding how to identify rational numbers helps in various mathematical situations, such as measuring, calculating proportions, and even in fields like engineering or science where precise values are necessary. Recognizing that a number is rational gives insights into its properties and how it might be used in calculations.