To solve the inequality \( x^{13} < 45 \), we can take the 13th root of both sides:
\[ x < 45^{1/13} \]
Now, let's calculate \( 45^{1/13} \):
- \( 45 \) can be approximated for easier calculation. Typically, we use a calculator for such an operation directly, but to get a rough estimate we can look at simple powers:
- Between \( 2^1 \) and \( 2^2 \) (which is \( 2 \) and \( 4 \)),
- and look at \( 3^{13} \) and \( 4^{13} \) (to find bounds).
Using a calculator: \[ 45^{1/13} \approx 1.63 \]
So the solution to \( x^{13} < 45 \) is:
\[ x < 45^{1/13} \approx 1.63 \]
Now, since none of the specific options given matches this, let’s analyze the responses to see if they're conceptually aligned, even if they don't yield direct numeric values close to 1.63:
- \( x < 1025 \)
- \( x < 10 \frac{2}{5} , \text{(which is } 10.4) \)
- \( x > 525 \)
- \( x < \frac{52}{5} , \text{(which is } 10.4) \)
- \( x < \frac{4}{65} \)
- \( x < 1345 \)
Since the correct side of the inequality is \( x < 1.63 \), we conclude:
None of the choices given accurately reflect this derived inequality.
For those who might be looking for a numerical estimation, the only viable option that remains conceptually consistent without providing any further approximation below 1.63 is essentially the understanding of polynomial root leads.
Thus no option fits perfectly, but the analytic approach suggests: The correct numerical solution is \( x < 45^{1/13} \text{ or } 1.63 \) which isn't listed directly.
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