Problem 22 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ -\sqrt[3]{125} $$

Step-by-Step Solution

Answer

The evaluated expression is -5.

Step 1: Identify the Cube Root

First, recognize that the expression involves finding a cube root. The cube root symbol \( \sqrt[3]{} \) indicates that we need to find a number which, when multiplied by itself three times, gives 125. Thus, we need to find \( \sqrt[3]{125} \).

Step 2: Determine the Cube Root of 125

Consider the cube of smaller positive integers sequentially. Calculate \( 5 \times 5 \times 5 = 125 \). Therefore, \( \sqrt[3]{125} = 5 \) as 5 cubed equals 125.

Step 3: Apply the Negative Sign

The expression is \( -\sqrt[3]{125} \). Since we have determined \( \sqrt[3]{125} = 5 \), we apply the negative sign to this result. Therefore, \( -\sqrt[3]{125} = -5 \).

Key Concepts

Understanding Real Numbers

Real numbers include both rational and irrational numbers. They encompass all the numbers that can be represented on the number line. This means numbers like fractions, whole numbers, and even decimals.
Real numbers include:

• Natural numbers (e.g., 1, 2, 3...)
• Whole numbers (e.g., 0, 1, 2, 3...)
• Integers (e.g., -3, -2, -1, 0, 1...)
• Rational numbers (e.g., 1/2, 3.7)
• Irrational numbers (e.g., \( \pi \), \( \sqrt{2} \))

They can be positive, negative, or zero. When dealing with operations like finding cube roots, it’s important to understand how these various numbers interact under the real number system. Anything you can think of placing on a number line falls under this category, including the root operations.

Cube Root Calculation

The cube root of a number is a special value that, when used three times in a multiplication, gives the original number. It’s like asking "What number cubed equals this?" In mathematical notation, the cube root of a number \( a \) is represented as \( \sqrt[3]{a} \).
For example, for 125, to find \( \sqrt[3]{125} \), you look for a number that when multiplied by itself twice more gives 125. By testing smaller numbers, you’ll find:

• \( 5 \times 5 \times 5 = 125 \)

Therefore, \( \sqrt[3]{125} = 5 \). It's important to systematically check numbers or use factors to make this determination easily.

Negative Sign Application

Applying negative signs in mathematical expressions is an important step that affects the result. In the original exercise, after calculating the cube root, we must then apply a negative sign to that result, because the original expression had a negative sign in front of the cube root symbol.
This step changes the outcome from positive to negative:

• If \( \sqrt[3]{125} = 5 \), then the original expression \( -\sqrt[3]{125} = -5 \).

It's crucial to remember that the negative sign outside the root applies only after you resolve the cube root. Applying it correctly will ensure your final answer reflects the negative value as intended.