In Exercises 1–38, multiply as indicated. If possible, simplify any radical expressions that appear in the product.³√x (³√24x² - ³√x)

Verified step by step guidance

1. Identify the expression to be multiplied: \( \sqrt[3]{x} (\sqrt[3]{24x^2} - \sqrt[3]{x}) \).
2. Distribute \( \sqrt[3]{x} \) to each term inside the parentheses.
3. Multiply \( \sqrt[3]{x} \) by \( \sqrt[3]{24x^2} \) using the property \( \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab} \).
4. Multiply \( \sqrt[3]{x} \) by \( \sqrt[3]{x} \) using the same property, resulting in \( \sqrt[3]{x^2} \).
5. Combine the results from the distribution and simplify any radical expressions if possible.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. In this context, the cube root (³√) indicates the value that, when raised to the third power, gives the original number. Understanding how to manipulate these expressions is crucial for simplifying and multiplying them effectively.

Multiplication of Radicals

When multiplying radical expressions, you can combine the radicands (the numbers inside the radical) under a single radical sign. For example, ³√a * ³√b = ³√(a*b). This property is essential for simplifying the product of radicals, as seen in the given expression.

Simplifying Radical Expressions

Simplifying radical expressions involves reducing them to their simplest form, which may include factoring out perfect cubes or squares. This process often requires identifying factors of the radicand that can be expressed as a whole number outside the radical, making the expression easier to work with and understand.