Simplify each radical. Assume all variables represent positive real numbers. ⁶√11³

Verified step by step guidance

1. Identify the expression: \( \sqrt[6]{11^3} \).
2. Recognize that this is a radical expression with a fractional exponent: \( 11^{3/6} \).
3. Simplify the fractional exponent: \( 3/6 = 1/2 \).
4. Rewrite the expression using the simplified exponent: \( 11^{1/2} \).
5. Express \( 11^{1/2} \) as a radical: \( \sqrt{11} \).

Key Concepts

Radical Expressions

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

Properties of Exponents

The properties of exponents govern how to simplify expressions involving powers. For instance, the rule a^(m/n) = n√(a^m) allows us to express roots in terms of exponents. This is essential for simplifying radical expressions, as it helps convert between radical and exponential forms.

Simplification of Radicals

Simplifying radicals involves reducing them to their simplest form, which often includes factoring out perfect squares or cubes. For example, when simplifying ⁶√11³, one must identify how many times the base can be expressed as a perfect sixth power, allowing for easier computation and clearer results.