Perform the indicated operations. Assume all variables represent positive real numbers. $3x\(\sqrt\)[3]{xy^2} - 2\(\sqrt\)[3]{8x^4y^2}$

Verified step by step guidance

1. Identify the terms to be operated on: \(3x\sqrt[3]{xy^{2}}\) and \(2\sqrt[3]{8x^{4}y^{2}}\).
2. Rewrite the cube roots as fractional exponents: \(\sqrt[3]{xy^{2}} = (xy^{2})^{\frac{1}{3}}\) and \(\sqrt[3]{8x^{4}y^{2}} = (8x^{4}y^{2})^{\frac{1}{3}}\).
3. Simplify the second cube root by factoring inside the root: \$8 = 2^{3}$, so \(\sqrt[3]{8x^{4}y^{2}} = \sqrt[3]{2^{3} \cdot x^{4} \cdot y^{2}}\).
4. Apply the cube root to each factor separately: \(\sqrt[3]{2^{3}} = 2\), \(\sqrt[3]{x^{4}} = x^{\frac{4}{3}}\), and \(\sqrt[3]{y^{2}} = y^{\frac{2}{3}}\).
5. Rewrite the original expression using these simplifications and then combine like terms by expressing all parts with fractional exponents to facilitate addition or subtraction.

Key Concepts

Radical Expressions and Cube Roots

Radical expressions involve roots such as square roots or cube roots. The cube root of a number x, denoted ∛x, is a value that when cubed gives x. Understanding how to simplify and manipulate cube roots is essential for combining terms involving radicals.

Properties of Exponents

Exponents indicate repeated multiplication. When working with radicals, exponents can be manipulated using rules like multiplying powers, dividing powers, and converting between radicals and fractional exponents, which helps simplify expressions under the root.

Combining Like Terms with Radicals

To perform operations like addition or subtraction on radical expressions, the terms must have the same radicand and root degree. Simplifying each radical to its simplest form allows identification of like terms, enabling proper combination of coefficients.