Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ⁵√x² • ⁵√x³

Verified step by step guidance

1. Identify the expression: \( \sqrt[5]{x^2} \cdot \sqrt[5]{x^3} \).
2. Use the property of radicals: \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \).
3. Combine the expressions under a single radical: \( \sqrt[5]{x^2 \cdot x^3} \).
4. Simplify the expression inside the radical: \( x^2 \cdot x^3 = x^{2+3} = x^5 \).
5. Apply the rule \( \sqrt[n]{a^n} = a \) to simplify \( \sqrt[5]{x^5} \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. In this case, we are dealing with fifth roots, denoted as ⁵√. Understanding how to manipulate these expressions is crucial, as it allows us to simplify and combine them effectively, especially when they share the same index.

Properties of Exponents

The properties of exponents are fundamental rules that govern how to handle expressions involving powers. For instance, when multiplying like bases, you add the exponents. This concept is essential for simplifying the product of radical expressions, as it allows us to express the radicals in terms of exponents, making calculations more straightforward.

Combining Radicals

Combining radicals involves using the properties of radicals to simplify expressions. When multiplying radicals with the same index, you can multiply the radicands (the numbers or expressions inside the radical) together. This principle is key to solving the given problem, as it enables the simplification of the product of the two fifth roots into a single radical expression.