Problem 24 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt[3]{-8 y^{3}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-2y\).

Step 1: Identify the Cube Root of Each Component

The expression is \( \sqrt[3]{-8y^3} \). Identify the cube root of the constant, \(-8\), and the variable, \(y^3\). The number \(-8\) is the cube of \(-2\), and \(y^3\) is the cube of \(y\).

Step 2: Apply the Cube Root to Each Component

The cube root of \(-8\) is \(-2\), since \((-2)^3 = -8\). The cube root of \(y^3\) is \(y\), because \(y^3\) is the cube of \(y\). Hence, \( \sqrt[3]{y^3} = y \).

Step 3: Simplify the Expression

Combine the results of the previous step to get the simplified form of the expression. Since \( \sqrt[3]{-8} = -2 \) and \( \sqrt[3]{y^3} = y \), the expression simplifies to \(-2y\).

Key Concepts

Simplifying Expressions

When we simplify expressions, we aim to rewrite them in their simplest form, while maintaining their original value. Simplification allows us to work with expressions more efficiently and makes it easier to analyze their properties. In the context of cube roots, we break down the components of an expression individually and apply the cube root operation.

For this exercise, the expression is \(\sqrt[3]{-8 y^{3}} \).

• First, identify the cube roots of individual components separately.
• The cube root of \(-8\) is \(-2\), since \((-2)^3 = -8\). This reflects the fundamental idea of a cube root: finding a number that, when multiplied by itself three times, equals the original number.
• Similarly, the cube root of \(y^3\) is \(y\), because \((y)^3 = y^3\).

Once each part is simplified, we recombine the results to get the final expression: \(-2y\).

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators such as addition and multiplication. They form a core part of algebra, allowing us to generalize and manipulate mathematical statements.

In the problem we are discussing, \( \sqrt[3]{-8 y^{3}} \), we combine constants and variables under a cube root.

• Algebraic expressions can include operations like exponents and roots, which we can simplify or rearrange following specific rules.
• Variables, such as \(y\) in this case, represent unknown quantities and are manipulated using algebraic rules.
• By applying known rules (like the rules for cube roots), we determine the simplest form of expression.

This skill is crucial for solving equations and understanding more complex mathematical structures.

Mathematics Education

Mathematics education focuses on developing skills that enable learners to understand and apply mathematical concepts. Simplifying expressions is a key skill in this process, promoting problem-solving and logical thinking.

Cube roots can sometimes be a tricky concept, but with practice, students can learn how to handle them efficiently.

• One objective in learning mathematics is to break down complicated problems into simpler, more manageable parts.
• By practicing simplification, learners develop familiarity with mathematical operations and gain confidence in their abilities.
• Educators often emphasize understanding why each step in a simplification process is taken, reinforcing the rules and patterns in mathematics.

Building these foundational skills prepares students for future studies in mathematics, ensuring they have the tools needed to tackle more advanced problems.