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

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Identify the expression inside the radical: the 9th root of 5 cubed, which can be written as $\sqrt[9]{5^{3}}$.

Recall the property of radicals that $\sqrt[n]{a^{m}} = a^{\frac{m}{n}}$. Apply this to rewrite the expression as $5^{\frac{3}{9}}$.

Simplify the fraction in the exponent: $\frac{3}{9}$ reduces to $\frac{1}{3}$, so the expression becomes $5^{\frac{1}{3}}$.

Recognize that $5^{\frac{1}{3}}$ is the cube root of 5, which is written as $\sqrt[3]{5}$.

Therefore, the simplified form of $\sqrt[9]{5^{3}}$ is $\sqrt[3]{5}$.

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Key Concepts

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

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.

Simplify each radical. Assume all variables represent positive real numbers. 539\(\sqrt\)[9]{5^3} 953

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Key Concepts

Radical Expressions and Roots

Simplifying Radicals Using Exponent Rules

Assumption of Positive Variables

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