Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ∛ 7x • ∛ 2y

Verified step by step guidance

1. Recall the rule for multiplying radicals with the same index: \( \sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{a \times b} \). In this problem, both radicals are cube roots (index 3).
2. Write the product of the cube roots as a single cube root: \( \sqrt[3]{7x} \times \sqrt[3]{2y} = \sqrt[3]{(7x)(2y)} \).
3. Multiply the expressions inside the cube root: \( (7x)(2y) = 14xy \).
4. Rewrite the expression as \( \sqrt[3]{14xy} \), which is the simplified form of the product.
5. Since all variables represent positive real numbers, no further simplification or absolute value considerations are needed.

Key Concepts

Properties of Radicals

Radicals represent roots of numbers or expressions. The cube root (∛) of a number is the value that, when cubed, returns the original number. Understanding how to manipulate radicals, especially nth roots, is essential for simplifying and performing operations with them.

Multiplication of Radicals with the Same Index

When multiplying radicals that have the same root index, you can multiply the radicands (the expressions inside the radical) together under a single radical. For example, ∛a × ∛b = ∛(a × b). This property simplifies the multiplication of radical expressions.

Assumption of Positive Real Numbers

Assuming all variables represent positive real numbers allows us to avoid considering absolute values or complex numbers when simplifying radicals. This assumption ensures that the operations and simplifications follow standard rules without additional restrictions.