Problem 29 - College Algebra with Modeling and Visualization

Question

If possible, simplify the expression by hand. If you cannot, approximate the answer to the nearest hundredth. Variables represent any real number. $$ \sqrt[3]{-64} $$

Step-by-Step Solution

Answer

The cube root of \\(-64\\) is \\-4\\.

Step 1: Identify the Cube Root

We are tasked with finding the cube root of \(-64\). A cube root is a number that when multiplied by itself twice (three times total), gives the original number. We are looking for a number \(x\) such that \(x^3 = -64\).

Step 2: Find the Cube Root of a Negative Number

When dealing with cube roots, negative numbers are straightforward because \( (-a)^3 = -(a^3) \) for any real number \(a\). Thus, the cube root of a negative number remains negative. We need a number, say \(-a\), such that \((-a)^3 = -64\).

Step 3: Calculate the Cube Root

Since \( (4)^3 = 64\), then \((-4)^3 = -64\). So, the cube root of \(-64\) is \-4\.

Key Concepts

Negative Numbers and Cube Roots

Negative numbers often confuse students because of the different rules they follow. When working with numbers, it's vital to know how negative numbers behave under different operations. In the context of cube roots, a negative result emerges when we have a negative input. This is different from square roots, where negative numbers don't have real roots. A cube root of a number \(x\) is such that \(x^3 = a\). If \(a = -64\), the cube root results in \(-4\) as \((-4)^3 = -64\). Therefore, negative numbers remain negative after taking the cube root.

Simplifying Expressions

Simplifying expressions generally means breaking them down to their simplest form. This helps us find a more straightforward way of expressing a mathematical idea. In the case of \(\sqrt[3]{-64}\), we broke down the problem by recognizing the cube root property. Here, instead of approximating the cube root, we simplified by identifying that

• \( (4)^3 = 64 \) which implies
• \( (-4)^3 = -64 \)

Thus, knowing the perfect cube helps simplify the expression instantly to \(-4\). Simplifying expressions is crucial in algebra to make complex problems more manageable.

Algebraic Solutions and Understanding

In algebra, solutions often rely on understanding the properties of operations and how they apply to different types of numbers, such as positives, negatives, integers, and real numbers. The key to solving cube root problems algebraically is recognizing patterns in numbers. For instance, knowing how to reorganize and simplify an expression helps solve it efficiently. Understanding exponential relationships, like those in cube roots, is an algebraic tool. In the exercise of cube roots:

• Identify the base of the operation - knowing that the cube root reverses cubing.
• Apply negative properties where applicable, leading to solutions for negative roots.

Therefore, recognizing these foundations allows for algebraic solutions that ensure clarity and sharpen problem-solving skills.