Simplify each radical. Assume all variables represent positive real numbers. $\(\sqrt\)[9]{5^3}$

Verified step by step guidance

1. Identify the expression inside the radical: the 9th root of 5 cubed, which can be written as \(\sqrt[9]{5^{3}}\).
2. Recall the property of radicals that \(\sqrt[n]{a^{m}} = a^{\frac{m}{n}}\). Apply this to rewrite the expression as \(5^{\frac{3}{9}}\).
3. Simplify the fraction in the exponent: \(\frac{3}{9}\) reduces to \(\frac{1}{3}\), so the expression becomes \(5^{\frac{1}{3}}\).
4. Recognize that \(5^{\frac{1}{3}}\) is the cube root of 5, which is written as \(\sqrt[3]{5}\).
5. Therefore, the simplified form of \(\sqrt[9]{5^{3}}\) is \(\sqrt[3]{5}\).

Key Concepts

Radical Expressions and Roots

A radical expression involves roots, such as square roots or nth roots. The nth root of a number is a value that, when raised to the nth power, gives the original number. For example, the 9th root of 5³ means finding a number which, when raised to the 9th power, equals 5³.

Simplifying Radicals Using Exponent Rules

Radicals can be rewritten using fractional exponents, where the nth root of a number is the same as raising that number to the power of 1/n. This allows simplification by multiplying exponents: (a^m)^(1/n) = a^(m/n). Applying this helps simplify expressions like ⁹√5³ to 5^(3/9).

Assumption of Positive Variables

Assuming all variables represent positive real numbers ensures that roots and exponents are well-defined and real-valued. This assumption avoids complications with negative bases or complex numbers, allowing straightforward simplification of radicals without considering absolute values or imaginary results.