Respuesta :
Answer:
To find the value of \(z^{2022} + \frac{1}{z^{2022}}\), we can use the given information about \(z + \frac{1}{z} = \sqrt{3}\).
Let's square the given equation \(z + \frac{1}{z} = \sqrt{3}\):
\[(z + \frac{1}{z})^2 = (\sqrt{3})^2\]
Simplify the left side:
\[z^2 + 2 + \frac{1}{z^2} = 3\]
Now, rearrange the equation:
\[z^2 + \frac{1}{z^2} = 3 - 2 = 1\]
Now, we have \(z^2 + \frac{1}{z^2} = 1\). To find \(z^{2022} + \frac{1}{z^{2022}}\), we can use the following relation:
\[z^{n} + \frac{1}{z^{n}} = (z + \frac{1}{z})^n\]
So,
\[z^{2022} + \frac{1}{z^{2022}} = (z + \frac{1}{z})^{2022}\]
Since \(z + \frac{1}{z} = \sqrt{3}\), substitute this into the equation:
\[(z + \frac{1}{z})^{2022} = (\sqrt{3})^{2022}\]
The result will be \((\sqrt{3})^{2022}\). However, if you want a numerical value, you can calculate \((\sqrt{3})^{2022}\). Keep in mind that this is a simplified expression, and the actual value will depend on the context of the problem.
