Simplify each radical. Assume all variables represent positive real numbers. ⁹√5³

Verified step by step guidance

1. Identify the expression: \( \sqrt[9]{5^3} \).
2. Recognize that the expression is a radical with a fractional exponent: \( 5^{\frac{3}{9}} \).
3. Simplify the fractional exponent: \( \frac{3}{9} = \frac{1}{3} \).
4. Rewrite the expression using the simplified exponent: \( 5^{\frac{1}{3}} \).
5. Express the result as a radical: \( \sqrt[3]{5} \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and higher-order roots. The notation ⁹√ indicates the ninth root of a number. Understanding how to manipulate these expressions is crucial for simplification, especially when dealing with exponents and variables.

Exponents and Roots

Exponents represent repeated multiplication, while roots are the inverse operation. For example, the expression 5³ means 5 multiplied by itself three times, and the ninth root of a number asks what number, when raised to the ninth power, equals that number. This relationship is essential for simplifying radical expressions.

Simplification of Radicals

Simplifying radicals involves reducing them to their simplest form, which often includes factoring out perfect squares or cubes. In the case of ⁹√5³, recognizing that 5³ can be expressed in terms of its prime factors helps in simplifying the radical expression effectively.