Perform the indicated operations. Assume all variables represent positive real numbers. $2\sqrt[3]{3} + 4\sqrt[3]{24} - \sqrt[3]{81}$

Verified step by step guidance

1. Identify the cube roots in the expression: \$2\sqrt[3]{3} + 4\sqrt[3]{24} - \sqrt[3]{81}$.
2. Simplify each cube root by factoring the radicand into prime factors and extracting perfect cubes: For example, \(\sqrt[3]{24} = \sqrt[3]{8 \times 3}\) and \(\sqrt[3]{81} = \sqrt[3]{27 \times 3}\).
3. Rewrite the expression using the simplified cube roots: \$2\sqrt[3]{3} + 4\sqrt[3]{8 \times 3} - \sqrt[3]{27 \times 3}$.
4. Extract the cube roots of the perfect cubes: \(\sqrt[3]{8} = 2\) and \(\sqrt[3]{27} = 3\), then rewrite the terms accordingly.
5. Combine like terms by factoring out the common cube root \(\sqrt[3]{3}\) and then perform the arithmetic operations on the coefficients.

Key Concepts

Simplifying Radicals

Simplifying radicals involves expressing the radicand (the number inside the root) as a product of perfect powers and other factors. This allows you to extract perfect cubes (for cube roots) outside the radical, making the expression easier to work with and combine.

Like Radicals and Combining Terms

Only radicals with the same index and radicand can be combined through addition or subtraction. After simplifying, identify like radicals to add or subtract their coefficients, similar to combining like terms in algebra.

Properties of Cube Roots

The cube root of a product equals the product of the cube roots: ∛(a·b) = ∛a · ∛b. This property helps break down complex radicands into simpler parts, facilitating simplification and combination of terms.