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 is of the form \((\sqrt[3]{a} - 1)(\sqrt[3]{a}^2 + \sqrt[3]{a} + 1)\), where \(a = 11\).
2. Recall the factorization identity for the difference of cubes: \((x - y)(x^2 + xy + y^2) = x^3 - y^3\).
3. Identify \(x = \sqrt[3]{11}\) and \(y = 1\) in the given expression.
4. Apply the difference of cubes formula: \((\sqrt[3]{11})^3 - 1^3\).
5. Simplify the cubes: \$11 - 1$.

Key Concepts

Properties of Radicals

Radicals represent roots of numbers, such as cube roots (∛). Understanding how to manipulate radicals, including simplifying and multiplying them, is essential. For cube roots, the product of ∛a and ∛b equals ∛(a*b), which helps in simplifying expressions involving radicals.

Multiplying Binomials and Polynomials

Multiplying expressions like (a - b)(c + d + e) requires applying the distributive property, multiplying each term in the first expression by each term in the second. This process expands the product into a sum of terms, which can then be combined or simplified.

Sum and Difference of Cubes Factorization

The expression (∛a - 1)(∛a² + ∛a + 1) is a factorization pattern for a - 1, based on the sum and difference of cubes formula: (x - y)(x² + xy + y²) = x³ - y³. Recognizing this pattern allows simplification of radical expressions by converting products into simpler forms.