Final answer:
To find the value of \(n\), the equation \(\sqrt{27} + \sqrt[n]{3} = 4 \times 3^{1/2}\) was simplified and solved, resulting in \(n = 2\) which is option A.
Step-by-step explanation:
The question asks to solve the equation \(\sqrt{27} + \sqrt[n]{3} = 4 \times 3^{1/2}\). First, we simplify the known terms. The square root of 27 can be simplified to \(3\sqrt{3}\), and \(4 \times 3^{1/2}\) simplifies to \(4\sqrt{3}\). Therefore, our equation can be rewritten as:
\(3\sqrt{3} + \sqrt[n]{3} = 4\sqrt{3}\)
To solve for \(n\), we need to isolate the term \(\sqrt[n]{3}\). Subtracting \(3\sqrt{3}\) from both sides, we get:
\(\sqrt[n]{3} = \sqrt{3}\)
We know that \(\sqrt{3}\) is the same as \(3^{1/2}\), so we want to find an \(n\) that makes \(\sqrt[n]{3}\) equal to \(3^{1/2}\). From the definition of radicals and exponents, we can see that the only option for \(n\) that meets this requirement is 2, because \(3^{1/n}\) would be equal to \(3^{1/2}\) if \(n = 2\). Thus, our solution is \(n = 2\), which corresponds to option A.