Perform the indicated operations. Assume all variables represent positive real numbers. (∛11 - 1) (∛11² + ∛11 +1)

Verified step by step guidance

1. Recognize that the expression \((\sqrt[3]{11} - 1)(\sqrt[3]{11^2} + \sqrt[3]{11} + 1)\) is a product of two binomials.
2. Recall the identity for the difference of cubes: \((a - b)(a^2 + ab + b^2) = a^3 - b^3\).
3. Identify \(a = \sqrt[3]{11}\) and \(b = 1\) in the expression.
4. Apply the difference of cubes identity: \((\sqrt[3]{11})^3 - 1^3\).
5. Simplify the expression to \(11 - 1\).

Key Concepts

Cube Roots

The cube root of a number 'x', denoted as ∛x, is a value 'y' such that y³ = x. Understanding cube roots is essential for simplifying expressions involving them, especially when performing operations like addition or multiplication. In this question, ∛11 and its powers are used, which requires familiarity with how to manipulate these roots.

Factoring Polynomials

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. The expression (∛11 - 1)(∛11² + ∛11 + 1) can be recognized as a product of a difference and a sum, which can be simplified using the identity for the difference of cubes. This concept is crucial for simplifying the given expression.

Properties of Exponents

Properties of exponents govern how to manipulate expressions involving powers. Key rules include the product of powers, power of a power, and the power of a product. In this problem, understanding how to apply these properties to cube roots and their powers is necessary for correctly performing the indicated operations and simplifying the expression.