Simplify each radical. Assume all variables represent positive real numbers. ⁹√∜7³

Verified step by step guidance

1. First, understand the expression: $\sqrt[9]{\sqrt[4]{7^3}}$. This is a nested radical expression.
2. Recognize that $\sqrt[4]{7^3}$ can be rewritten using fractional exponents as $(7^3{)}^{1/4}$.
3. Apply the power of a power property: $(a^m{)}^n=a^{m\cdot n}$. So, $(7^3{)}^{1/4}=7^{3/4}$.
4. Now, simplify the outer radical: $\sqrt[9]{7^{3/4}}$ can be rewritten as $(7^{3/4}{)}^{1/9}$.
5. Again, use the power of a power property: $(7^{3/4}{)}^{1/9}=7^{(3/4)\cdot (1/9)}=7^{3/36}=7^{1/12}$.

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, while '∜' indicates the fourth root. Understanding how to manipulate these expressions is crucial for simplification, especially when dealing with exponents and variables.

Properties of Exponents

The properties of exponents govern how to simplify expressions involving powers and roots. For instance, the rule that states a^(m/n) = n√(a^m) allows us to convert between radical and exponential forms. This is essential for simplifying expressions like ⁹√∜7³, as it helps in breaking down the roots into manageable parts.

Simplification of Radicals

Simplifying radicals involves reducing them to their simplest form, which often includes factoring out perfect squares, cubes, or higher powers. This process may also involve combining like terms and applying the properties of radicals. In the case of ⁹√∜7³, recognizing how to express the radical in terms of its prime factors is key to achieving a simplified result.