Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. $\(\sqrt\)[4]{\(\frac{g^3h^5}{9r^6}\)}$

Verified step by step guidance

1. Recognize that the expression involves a fourth root (∜) of a fraction: \(\sqrt[4]{\frac{g^{3} h^{5}}{9 r^{6}}}\).
2. Rewrite the fourth root of the fraction as the fraction of the fourth roots: \(\frac{\sqrt[4]{g^{3} h^{5}}}{\sqrt[4]{9 r^{6}}}\).
3. Apply the property of radicals to each variable inside the root by expressing the exponents as fractions: \(\sqrt[4]{g^{3}} = g^{\frac{3}{4}}\), \(\sqrt[4]{h^{5}} = h^{\frac{5}{4}}\), \(\sqrt[4]{9} = 9^{\frac{1}{4}}\), and \(\sqrt[4]{r^{6}} = r^{\frac{6}{4}}\).
4. Simplify the fractional exponents where possible, for example, \(r^{\frac{6}{4}}\) can be reduced to \(r^{\frac{3}{2}}\).
5. Combine all parts to write the simplified expression as \(\frac{g^{\frac{3}{4}} h^{\frac{5}{4}}}{9^{\frac{1}{4}} r^{\frac{3}{2}}}\).

Key Concepts

Radical Expressions and Roots

Radical expressions involve roots such as square roots, cube roots, and fourth roots (∜). The fourth root of a number is the value that, when raised to the fourth power, gives the original number. Understanding how to rewrite radicals as fractional exponents helps simplify expressions.

Properties of Exponents

Exponents indicate repeated multiplication, and their properties allow simplification of expressions. Key rules include multiplying powers with the same base by adding exponents, dividing by subtracting exponents, and raising a power to another power by multiplying exponents. These rules are essential when simplifying expressions under radicals.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing the numerator and denominator by factoring and canceling common terms. When variables have exponents, applying exponent rules before or after taking roots helps simplify the expression. Recognizing that variables represent positive real numbers allows ignoring absolute value considerations.