Simplify each radical. Assume all variables represent positive real numbers. ⁶√√5³

Verified step by step guidance

1. First, understand that the expression involves nested radicals: \( \sqrt[6]{\sqrt{5^3}} \).
2. Recognize that \( \sqrt{5^3} \) can be rewritten using fractional exponents as \( (5^3)^{1/2} \).
3. Apply the property of exponents \((a^m)^n = a^{m \cdot n}\) to simplify \( (5^3)^{1/2} \) to \( 5^{3/2} \).
4. Now, simplify the outer radical \( \sqrt[6]{5^{3/2}} \) by rewriting it as \( (5^{3/2})^{1/6} \).
5. Use the property of exponents again to combine the exponents: \( 5^{(3/2) \cdot (1/6)} \). Simplify the exponent to find the final expression.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and higher-order roots. The notation '√' indicates a square root, while '⁶√' denotes a sixth root. Understanding how to manipulate these expressions is crucial for simplification, especially when dealing with exponents and variables.

Properties of Exponents

The properties of exponents govern how to simplify expressions involving powers. Key rules include the product of powers, quotient of powers, and power of a power. For example, when simplifying radicals, one can convert roots into fractional exponents, which can then be manipulated using these properties.

Simplification of Radicals

Simplifying radicals involves reducing them to their simplest form, which often includes factoring out perfect squares or cubes. For instance, when simplifying '⁶√√5³', one must first express the radical in terms of exponents and then apply the properties of exponents to simplify the expression effectively.