Simplify.
$\sqrt[4]{80}+4 \sqrt[4]{405}$
To simplify the expression \(\sqrt[4]{80} + 4\sqrt[4]{405}\), we first analyze the terms within the radicals. We can simplify each radical by finding perfect fourth powers that divide the radicands. First, consider \(80\). Its prime factorization is \(80 = 16 \times 5 = 2^4 \times 5\). Thus, the fourth root is \(\sqrt[4]{80} = \sqrt[4]{2^4 \times 5} = 2\sqrt[4]{5}\). Next, consider \(405\). Its prime factorization is \(405 = 81 \times 5 = 3^4 \times 5\). Thus, the fourth root is \(\sqrt[4]{405} = \sqrt[4]{3^4 \times 5} = 3\sqrt[4]{5}\).
Now that we have simplified the individual radicals, we substitute these results back into the original expression: \[\sqrt[4]{80} + 4\sqrt[4]{405} = 2\sqrt[4]{5} + 4(3\sqrt[4]{5}).\] We can now simplify the second term: \(4(3\sqrt[4]{5}) = 12\sqrt[4]{5}\). Finally, combining the two terms yields: \[2\sqrt[4]{5} + 12\sqrt[4]{5} = 14\sqrt[4]{5}.\] The expression is now fully simplified.
Simplifying radical expressions is a fundamental skill in algebra that appears frequently in engineering and physics. For example, when calculating the diagonal distance of a complex shape or determining the intensity of sound or light over distance, intermediate steps often involve radical values that need to be simplified to obtain an exact and usable result. Being able to combine terms like \(2\sqrt[4]{5}\) and \(12\sqrt[4]{5}\) helps in maintaining precision throughout a multi-step calculation, preventing rounding errors early in the process.