Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 121

Chapter 1, Problem 121

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

Verified step by step guidance

Recognize that the expression is of the form $(\sqrt[3]{a} - 1)(\sqrt[3]{a}^2 + \sqrt[3]{a} + 1)$, where $a = 11$.

Recall the factorization identity for the difference of cubes: $(x - y)(x^2 + xy + y^2) = x^3 - y^3$.

Identify $x = \sqrt[3]{11}$ and $y = 1$ in the given expression.

Apply the difference of cubes formula: $(\sqrt[3]{11})^3 - 1^3$.

Simplify the cubes: $11 - 1$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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