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. Recognize that you are dealing with cube roots: \( \sqrt[3]{7x} \) and \( \sqrt[3]{2y} \).
2. Use the property of radicals that states \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \).
3. Apply this property to combine the cube roots: \( \sqrt[3]{7x} \cdot \sqrt[3]{2y} = \sqrt[3]{(7x) \cdot (2y)} \).
4. Multiply the expressions inside the cube root: \( 7x \cdot 2y = 14xy \).
5. Express the result as a single cube root: \( \sqrt[3]{14xy} \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. In this context, the cube root is denoted as ∛, which represents the value that, when multiplied by itself three times, gives the original number. Understanding how to manipulate these expressions is crucial for performing operations like addition, subtraction, multiplication, and division.

Multiplication of Radicals

When multiplying radical expressions, the product of the radicands (the numbers or expressions inside the radical) can be combined under a single radical. For example, ∛a • ∛b = ∛(a*b). This property simplifies the process of multiplying radicals and is essential for solving problems involving multiple radical terms.

Properties of Exponents

Understanding the properties of exponents is vital when working with radical expressions, as radicals can be expressed in exponential form. For instance, the cube root of a number can be written as that number raised to the power of 1/3. This knowledge allows for easier manipulation and simplification of expressions involving radicals, especially when combined with multiplication.