Simplify each radical. Assume all variables represent positive real numbers. $\(\sqrt\)[6]{11^3}$

Verified step by step guidance

1. Identify the expression to simplify: the sixth root of 11 cubed, written as \(\sqrt[6]{11^{3}}\).
2. Recall the property of radicals that allows rewriting the root as a fractional exponent: \(\sqrt[n]{a^{m}} = a^{\frac{m}{n}}\). Apply this to get \(11^{\frac{3}{6}}\).
3. Simplify the fractional exponent \(\frac{3}{6}\) by dividing numerator and denominator by their greatest common divisor, which is 3, resulting in \(\frac{1}{2}\).
4. Rewrite the expression with the simplified exponent: \(11^{\frac{1}{2}}\).
5. Recognize that \(11^{\frac{1}{2}}\) is equivalent to the square root of 11, or \(\sqrt{11}\), which is the simplified form of the original radical.

Key Concepts

Radical Expressions and Roots

A radical expression involves roots such as square roots, cube roots, or nth roots. The nth root of a number is a value that, when raised to the nth power, gives the original number. Understanding how to interpret and manipulate these roots is essential for simplifying radical expressions.

Properties of Exponents

Exponents represent repeated multiplication, and their properties allow us to rewrite and simplify expressions. For radicals, the nth root can be expressed as a fractional exponent (e.g., the sixth root as an exponent of 1/6), which helps in simplifying powers inside radicals by using exponent rules.

Simplifying Radicals Using Prime Factorization and Exponent Rules

Simplifying radicals often involves expressing the radicand as a product of prime factors or powers, then applying exponent rules to extract powers that match the root's index. For example, rewriting 11³ under a sixth root involves converting to fractional exponents and simplifying accordingly.