Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. $\(\frac{-4}{\sqrt[3]{3}\)} + \(\frac{1}{\sqrt[3]{24}\)} - \(\frac{2}{\sqrt[3]{81}\)}$

Verified step by step guidance

1. Identify the expression to simplify: \(\left(-\frac{4}{\sqrt[3]{3}}\right) + \left(\frac{1}{\sqrt[3]{24}}\right) - \left(\frac{2}{\sqrt[3]{81}}\right)\).
2. Rewrite each cube root in terms of prime factors or simpler cube roots if possible: For example, \(\sqrt[3]{24} = \sqrt[3]{8 \times 3} = \sqrt[3]{8} \times \sqrt[3]{3}\) and \(\sqrt[3]{81} = \sqrt[3]{3^4} = \sqrt[3]{3^3 \times 3} = 3 \times \sqrt[3]{3}\).
3. Express each term with the simplified cube roots: Replace \(\sqrt[3]{24}\) and \(\sqrt[3]{81}\) with their factored forms to have a common cube root base if possible.
4. Find a common denominator involving \(\sqrt[3]{3}\) to combine all terms into a single fraction. This may involve multiplying numerator and denominator of each term appropriately.
5. Combine the numerators over the common denominator and simplify the resulting expression by combining like terms.

Key Concepts

Cube Roots and Radicals

A cube root of a number is a value that, when multiplied by itself three times, gives the original number. It is denoted as ∛x. Understanding how to work with cube roots is essential for simplifying expressions involving radicals.

Simplifying Radicals

Simplifying radicals involves rewriting the radical expression in its simplest form by factoring out perfect cubes (for cube roots) or other powers. This process makes it easier to perform addition or subtraction of radical terms.

Adding and Subtracting Radical Expressions

To add or subtract radicals, the radicals must have the same radicand (the number inside the root) and index. If they differ, you must simplify or rewrite them to have a common radical form before combining the terms.