In Exercises 75–82, add or subtract terms whenever possible. ³√54xy³−y³√128x

Verified step by step guidance

1. First, identify the terms in the expression: \( \sqrt[3]{54xy^3} \) and \( -y\sqrt[3]{128x} \). Notice that both terms involve cube roots, so we will simplify each cube root separately.
2. Simplify \( \sqrt[3]{54xy^3} \): Break down 54 into its prime factors (\( 54 = 2 \cdot 3^3 \)), and separate the cube root of \( y^3 \) since it is a perfect cube. This gives \( \sqrt[3]{54xy^3} = 3y\sqrt[3]{2x} \).
3. Simplify \( -y\sqrt[3]{128x} \): Break down 128 into its prime factors (\( 128 = 2^7 \)), and extract the cube root of \( 2^6 \) (a perfect cube). This gives \( -y\sqrt[3]{128x} = -4y\sqrt[3]{2x} \).
4. Combine the simplified terms: Both terms now involve \( y\sqrt[3]{2x} \), so we can combine them by factoring out \( y\sqrt[3]{2x} \). This results in \( (3 - 4)y\sqrt[3]{2x} \).
5. Simplify the coefficient: Combine \( 3 - 4 \) to get \( -1 \), so the final expression is \( -y\sqrt[3]{2x} \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. In this question, we are dealing with cube roots, denoted as ³√. Understanding how to simplify these expressions is crucial, as it allows us to combine like terms effectively.

Like Terms

Like terms are terms that have the same variable parts raised to the same powers. For example, ³√54xy^3 and -y³√128x can be combined if they share the same radical component. Identifying like terms is essential for adding or subtracting expressions correctly.

Factoring and Simplifying

Factoring involves breaking down expressions into simpler components, which can help in simplifying radical expressions. For instance, recognizing that 54 and 128 can be factored into their prime components aids in simplifying the cube roots, making it easier to combine terms.