- Separate the radical into \(\sqrt[4]{81} \cdot \sqrt[4]{x^8} \cdot \sqrt[4]{y^5}\).
- Simplify the constant: \(\sqrt[4]{3^4} = 3\).
- Simplify the variable terms: \(\sqrt[4]{x^8} = x^2\) and \(\sqrt[4]{y^5} = y \sqrt[4]{y}\).
- Combine the terms to get \(\boxed{3 x^2 y \sqrt[4]{y}}\).
Explanation
- Analyzing the Expression
To simplify the expression \(\sqrt[4]{81 x^8 y^5}\), we can apply the properties of radicals which state that the nth root of a product is the product of the nth roots, i.e., \(\sqrt[n]{abc} = \sqrt[n]{a} \cdot \sqrt[n]{b} \cdot \sqrt[n]{c}\). We will treat the constant, the variable \(x\), and the variable \(y\) separately.
- Simplifying Each Component
First, we simplify the constant term \(\sqrt[4]{81}\). Since \(81 = 3^4\), we have \(\sqrt[4]{3^4} = 3\). Next, we simplify the variable term \(\sqrt[4]{x^8}\). Using the exponent rule \(\sqrt[n]{a^m} = a^{m/n}\), we get \(x^{8/4} = x^2\). Lastly, for the variable term \(\sqrt[4]{y^5}\), we can split the exponent into the largest multiple of 4: \(\sqrt[4]{y^4 \cdot y} = \sqrt[4]{y^4} \cdot \sqrt[4]{y} = y \cdot \sqrt[4]{y}\).
- Combining the Results
Now, we multiply the simplified parts together to obtain the final result:
\[3 \cdot x^2 \cdot y \cdot \sqrt[4]{y} = 3x^2y\sqrt[4]{y}\]
Comparing this result with the given options, we identify the second option as the correct simplification.
Examples
Radicals are essential in fields like physics and engineering, where we often work with power-based relationships. For instance, if you are calculating the side length of a cube given its volume or analyzing frequency-based wave propagation, simplifying expressions involving roots is a fundamental skill. Understanding how to reduce \(\sqrt[4]{81 x^8 y^5}\) to \(3x^2y\sqrt[4]{y}\) helps in simplifying complex physical formulas into manageable components for easier calculation and interpretation.