Problem 17 - College Algebra with Modeling and Visualization

Question

Find the cube root of the number. $$ -8 $$

Step-by-Step Solution

Answer

The cube root of -8 is -2.

Step 1: Understand the Problem

We need to find the cube root of -8, which means determining which number, when multiplied by itself three times, gives -8.

Step 2: Recall the Cube Root Concept

The cube root of a number \(x\) is a number \(y\) such that \(y^3 = x\). Therefore, we are looking for a number \(y\) where \(y^3 = -8\).

Step 3: Identify the Cube Root of -8

Knowing that \((-2)^3 = (-2) \times (-2) \times (-2) = -8\), we can determine that -2 is the cube root of -8.

Step 4: Verify the Solution

To ensure our solution is correct, we check: \((-2)^3 = (-2) \times (-2) \times (-2) = 4 \times (-2) = -8\). The calculation confirms that -2 is indeed the cube root of -8.

Key Concepts

Real Numbers

Real numbers comprise all the numbers you will likely encounter in everyday mathematics. They include positive numbers, negative numbers, zero, whole numbers, fractions, and decimals. Real numbers can be found on the number line, stretching infinitely in both the positive and negative directions.
Real numbers can be divided into two main categories:

• Rational numbers: These can be expressed as a fraction of two integers. Examples include 1/2, 4, and -6.
• Irrational numbers: These cannot be expressed as fractions, featuring non-repeating decimal parts. Common examples are π (pi) and √2.

In the context of cube roots, real numbers can yield both rational and irrational results. For instance, the cube root of 8 is 2, a rational number, while the cube root of some less neat numbers might be irrational.

Negative Numbers

Negative numbers are all the numbers that are less than zero. They are found to the left of zero on the number line. These numbers are crucial in understanding the full set of real numbers.
When dealing with cube roots, especially for negative numbers, the process is a bit different than dealing with square roots. Why? Because multiplying a negative number by itself an odd number of times still results in a negative number. Therefore, the cube root of a negative number is also negative. For example,

• The cube root of -8 is -2.
• This is because \((-2) \times (-2) \times (-2) = -8\).

Negative numbers, when paired with odd exponents (like 3 in cube roots), work quite smoothly in mathematical operations.

Exponents

Exponents are used to express repeated multiplication. When you see a number with an exponent, it means that the number is multiplied by itself a certain number of times. For example,

• The notation \(y^3\) indicates that \(y\) is multiplied by itself three times, which is \(y \times y \times y\).
• Cube roots then take this idea and reverse it: given \(y^3 = x\), you are finding a number \(y\) such that this multiplication results in \(x\).

When working with exponents, it’s important to remember:

• Even exponents (e.g., squared or raised to the fourth power) of real numbers always give a positive result.
• Odd exponents keep the sign of the base number; hence, \((-2)^3 = -8\) while \((-2)^2 = 4\).

Understanding these power rules is essential for finding cube roots, especially when dealing with negative numbers.