Perform the indicated operations. Assume all variables represent positive real numbers. 3x∛xy² - 2∛8x⁴y²

Verified step by step guidance

1. Identify the cube roots in the expression: \(3x\sqrt[3]{xy^2} - 2\sqrt[3]{8x^4y^2}\).
2. Simplify the cube root \(\sqrt[3]{8x^4y^2}\) by breaking it down: \(\sqrt[3]{8} \cdot \sqrt[3]{x^4} \cdot \sqrt[3]{y^2}\).
3. Calculate \(\sqrt[3]{8}\), which is 2, and simplify \(\sqrt[3]{x^4}\) to \(x^{4/3}\) and \(\sqrt[3]{y^2}\) to \(y^{2/3}\).
4. Substitute the simplified terms back into the expression: \(3x\sqrt[3]{xy^2} - 2 \cdot 2 \cdot x^{4/3}y^{2/3}\).
5. Simplify the expression further by combining like terms and simplifying coefficients.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. In this question, the cube root is denoted by the symbol ∛. Understanding how to manipulate these expressions, including simplifying and combining them, is essential for performing the indicated operations.

Properties of Exponents

Properties of exponents govern how to handle expressions involving powers. Key rules include the product of powers, power of a power, and the power of a product. These rules are crucial for simplifying terms like x⁴ and y² when combined with radical expressions.

Combining Like Terms

Combining like terms is a fundamental algebraic skill that involves adding or subtracting terms with the same variable parts. In this problem, recognizing which terms can be combined after simplification is key to arriving at the final answer. This requires a clear understanding of the coefficients and the variables involved.