Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. $\sqrt[4]{\frac{m}{n^4}}$

Verified step by step guidance

1. Identify the given expression: the fourth root of \( \frac{m}{n^4} \), which can be written as \( \sqrt[4]{\frac{m}{n^4}} \).
2. Recall the property of radicals that \( \sqrt[4]{\frac{a}{b}} = \frac{\sqrt[4]{a}}{\sqrt[4]{b}} \). Apply this to separate the numerator and denominator inside the radical: \( \frac{\sqrt[4]{m}}{\sqrt[4]{n^4}} \).
3. Use the rule that \( \sqrt[4]{n^4} = n^{\frac{4}{4}} = n^1 = n \), since the fourth root and the fourth power cancel each other out.
4. Rewrite the expression as \( \frac{\sqrt[4]{m}}{n} \).
5. This is the simplified form using the rules for radicals, assuming all variables represent positive real numbers.

Key Concepts

Radical Expressions and Their Properties

Radical expressions involve roots such as square roots, cube roots, and fourth roots. Understanding how to simplify and manipulate these expressions is essential, including recognizing that the nth root of a quotient equals the quotient of the nth roots.

Rules for Operations with Radicals

Operations with radicals follow specific rules, such as the product rule (√a * √b = √(ab)) and the quotient rule (√(a/b) = √a / √b). Applying these rules allows simplification of expressions involving radicals, especially when variables and exponents are involved.

Exponents and Their Relationship to Radicals

Radicals can be expressed as fractional exponents, where the nth root of a variable is the variable raised to the 1/n power. This connection helps in simplifying and performing operations on radical expressions by converting between radical and exponential forms.