Problem 21 - 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

\(-5\) is the cube root of \(-125\).

Step 1: Understand the Cube Root

To evaluate \( \sqrt[3]{-125} \), we need to understand that the cube root of a number \( x \) is a number \( y \) such that \( y^3 = x \). In this case, we need to find a number whose cube is \(-125\).

Step 2: Identify the Cube

Recognize that \(-125\) is the cube of a number. A number multiplied by itself three times should give \(-125\).

Step 3: Calculate the Cube Root

Calculate manually or by intuition that \(-5\) when cubed yields \(-125\). Therefore, the cube root of \(-125\) is \(-5\). Check: \( (-5) \times (-5) \times (-5) = -125 \).

Step 4: Reconfirm the Calculation

Verify the calculation: 1. \( (-5) \times (-5) = 25 \).2. \( 25 \times (-5) = -125 \).Thus, the calculation confirms that \((-5)^3 = -125\).

Key Concepts

Real Numbers

Real numbers include both rational and irrational numbers. These can be any value that exists along the number line, which means it includes both positive and negative numbers, and zero as well.
The set of real numbers is denoted by the symbol \( \mathbb{R} \). This broad set provides a context for various arithmetic operations and functions, such as square roots and cube roots.

• Rational numbers are values that can be written as a fraction, where both the numerator and the denominator are integers and the denominator is not zero.
• Irrational numbers cannot be expressed as a simple fraction, such as \( \pi \) or \( \sqrt{2} \).

Real numbers are fundamental in solving equations since they help us find values that satisfy certain conditions like having a particular root, in this case, a cube root of \( -125 \).

Negative Numbers

Negative numbers are numbers less than zero. Think of them as the opposite of positive numbers. They appear on the left side of zero on a number line. Negative numbers are usually denoted by a minus sign (-) before the number.
They play a crucial role in calculations involving subtraction, debt, or temperature drops, among other real-life applications.

• In the realm of cube roots, finding the cube root of a negative number involves identifying a negative number that, when raised to the power of three, results in the given negative value. For example, the cube root of \(-125\) is \(-5\) because \((-5) \times (-5) \times (-5) = -125\).

Understanding negative numbers is important because operations involving them follow different rules than those involving positive numbers.

Exponentiation

Exponentiation is a mathematical operation that involves raising a base number to a power. When raising a number to an exponent, you're essentially multiplying the base by itself repeatedly, as many times as the exponent indicates.
For example, the expression \(y^n\) means \(y\) is multiplied by itself \(n\) times.

• The element "cube" in cube roots directly relates to exponentiation, specifically the power of three.
• When finding a cube root, you are essentially finding a number which, when cubed, returns to the original number.

In our problem of evaluating \( \sqrt[3]{-125} \), we find \(-5\) is the number that satisfies \(y^3 = -125\). This means \((-5)^3\) returns \(-125\), demonstrating exponentiation in reverse.

Manual Calculation

Manual calculation is a method by which mathematical problems are solved without the aid of a calculator or computer software. This approach requires understanding the properties of numbers and applying logical reasoning.
Performing manual calculations helps improve mathematical intuition and problem-solving skills.

• For example, to manually calculate \( \sqrt[3]{-125} \), we determine a number \((-5)\) which, when cubed, equals \(-125\).
• This process involves breaking down calculations, such as proving \((-5) \times (-5) = 25\) and then until it becomes \(25 \times (-5) = -125\).
• Approaching problems manually reinforces fundamental arithmetic operations.

Manual calculations ensure that students grasp each concept's theoretical underpinning, confidently tackling similar mathematical problems in varied contexts.