Problem 58 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt[3]{-125} $$

Step-by-Step Solution

Answer

The simplified expression is \(-5\).

Step 1: Identify the Expression

Recognize that the expression given is \( \sqrt[3]{-125} \). This represents the cube root of \(-125\).

Step 2: Break Down the Components

Notice that \(-125\) can be expressed as \(-1 \times 125\). Since the cube root distributes over multiplication, we can think of it as \(\sqrt[3]{-1} \times \sqrt[3]{125}\).

Step 3: Evaluate Cube Root of -1

The cube root of \(-1\) is simply \(-1\), because \((-1)^3 = -1\). Thus, \(\sqrt[3]{-1} = -1\).

Step 4: Evaluate Cube Root of 125

Recognize that \(125 = 5^3\). Therefore, \(\sqrt[3]{125} = 5\), as the cube root of \(5^3\) is \(5\).

Step 5: Combine the Results

Multiply the results from Step 3 and Step 4: \(-1 \times 5 = -5\). Thus, \(\sqrt[3]{-125} = -5\).

Key Concepts

Simplifying Expressions

When dealing with mathematical expressions, the main goal is often to simplify them into a more manageable form. Simplifying expressions involves reducing the expression to its simplest form without changing its value. In the context of cube roots, like with the given example \( \sqrt[3]{-125} \), we aim to express this in the simplest form.

• Identifying parts is crucial: Break complex numbers down into prime factors.
• Use properties of powers and roots to simplify: For cube roots, remember that \( a^3 \) becomes \( a \) when under a cube root.

Recognizing patterns like perfect cubes helps, as it allows for an immediate simplification. By evaluating both positive and negative numbers, and using operations like subtraction and negation, we get the simplest possible outcome.

Rationalizing the Denominator

Rationalizing the denominator refers to the process of eliminating radicals from the denominator of a fraction. Although this primarily applies to square roots, it's an essential concept to understand in simplifying expressions involving roots of any kind.

• Ensure no radicals remain in the denominator, as rational numbers are easier to work with.
• Multiply by a form of 1, such as \( \frac{\sqrt[3]{a}}{\sqrt[3]{a}} \), to eliminate radicals.

In the expression \( \sqrt[3]{-125} \), this concept is information for reducing down to \( -5 \). While the focus is not directly on the denominator here, understanding how to handle and simplify under root radical operations ensures clearer algebraic expressions.

Negative Numbers

Understanding how negative numbers interact within expressions, especially those involving roots, is critical. In our exercise, the expression \( \sqrt[3]{-125} \) incorporates a negative number under a cube root.

• Note that odd roots (like cube roots) of negative numbers still yield negative results.
• Cube root of \( -1 \) is \( -1 \), unlike square roots where negative numbers create imaginary numbers.

So, effectively solving roots of negative numbers requires a keen understanding of these properties. Using this knowledge, we can handle negative numbers gracefully within mathematical operations.

Multiplication of Roots

Multiplication of roots is a common operation in simplifying radical expressions. In the given example \( \sqrt[3]{-1} \times \sqrt[3]{125} \), we use this operation:

• Cube roots distribute over multiplication, similar to exponents. Thus, we can break \( \sqrt[3]{-125} \) into separate roots.
• Multiply the simplified individual roots to combine them into one product.

This simplifies the process and helps to avoid errors. Always ensure clarity by performing operations step-by-step, confirming each multiplication's validity. In our solved example, this results in the neat outcome: \( -1 \times 5 = -5 \).