The given equation appears to involve a relationship between \( a^n \), and from the way it is written, it seems to indicate a pattern based on the values of \( n \).
Let's interpret the equation \( a^n = 5 + 2 \times (n = 1) \):
- The expression \( (n = 1) \) evaluates to 1 when \( n \) is 1, and 0 otherwise.
- Therefore, when \( n = 1 \): \[ a^1 = 5 + 2 \times 1 = 5 + 2 = 7 \quad \Rightarrow \quad a = 7 \]
- For \( n = 2 \): \[ a^2 = 5 + 2 \times 0 = 5 + 0 = 5 \quad \Rightarrow \quad a^2 = 5 \quad \Rightarrow \quad a = \sqrt{5} \]
- For other values of \( n \), the term \( (n = 1) \) would be 0, hence: \[ a^n = 5 \] for \( n = 3, 4, 5, \ldots \) which gives \( a = \sqrt[n]{5} \).
Now, to solve for \( x \) and \( y \):
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If \( x = 2 \), that means \( n = 2 \): \[ a^2 = 5 \quad \Rightarrow \quad y = 5 \]
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If \( x = 4 \), that means \( n = 4 \): \[ a^4 = 5 \quad \Rightarrow \quad y = 5 \]
Therefore, we can summarize the results as follows:
| x | y | |---|----| | 2 | 5 | | 4 | 5 |
So for the values provided:
- When \( x = 2 \), \( y = 5 \)
- When \( x = 4 \), \( y = 5 \)