Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ⁵√∛9

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Identify the expression: the fifth root of the cube root of 9, which can be written as \(\sqrt[5]{\sqrt[3]{9}}\).

Rewrite the nested radicals using rational exponents: \(\sqrt[3]{9} = 9^{\frac{1}{3}}\), so the expression becomes \(\sqrt[5]{9^{\frac{1}{3}}}\).

Apply the rule that \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\), so \(\sqrt[5]{9^{\frac{1}{3}}} = 9^{\frac{1}{3} \times \frac{1}{5}}\).

Multiply the exponents: \(\frac{1}{3} \times \frac{1}{5} = \frac{1}{15}\), so the expression simplifies to \(9^{\frac{1}{15}}\).

Express the final answer as a single radical or rational exponent: \(9^{\frac{1}{15}}\) represents the 15th root of 9.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Radical Expressions and Their Properties

Radical expressions involve roots such as square roots, cube roots, and higher-order roots. Understanding how to interpret and manipulate these roots is essential, including recognizing that the nth root of a number is the value that, when raised to the nth power, returns the original number.

Converting Radicals to Rational Exponents

Radicals can be expressed as rational exponents, where the nth root of a number is written as that number raised to the power of 1/n. This conversion simplifies operations like multiplication and division of radicals by allowing the use of exponent rules.

Rules for Multiplying and Simplifying Radicals

When multiplying radicals with the same index, you can multiply the radicands (the numbers inside the roots) and keep the same root. Simplifying involves combining like terms and reducing the expression to its simplest radical form, assuming all variables represent positive real numbers.

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