To complete the table, we'll find the values of \( x^{-2} \), \( x^{-3} \), \( x^{-4} \), \( x^{-13} \), and \( x^{-1} \).
Recall that \( x^{-n} = \frac{1}{x^n} \).
Given Problems and Their Values:
- \( 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \)
- \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \)
- \( 10^{-1} = \frac{1}{10^1} = \frac{1}{10} \)
For \( x \):
The values of \( x^{-n} \) can be expressed in terms of \( \frac{1}{x^n} \). Let's summarize the computations:
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\( x^{-2} \): \[ x^{-2} = \frac{1}{x^2} \]
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\( x^{-3} \): \[ x^{-3} = \frac{1}{x^3} \]
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\( x^{-4} \): \[ x^{-4} = \frac{1}{x^4} \]
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\( x^{-13} \): \[ x^{-13} = \frac{1}{x^{13}} \]
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\( x^{-1} \): \[ x^{-1} = \frac{1}{x} \]
Completing the Table:
Now, we can denote the values for the columns:
- \( x^{-2} = \frac{1}{x^2} \)
- \( x^{-3} = \frac{1}{x^3} \)
- \( x^{-4} = \frac{1}{x^4} \)
- \( x^{-13} = \frac{1}{x^{13}} \)
- \( x^{-1} = \frac{1}{x} \)
Since we don’t have specific values for \( x \), we cannot assign numerical values for these expressions without more information about \( x \).
Thus, the completed table looks like this:
| Problem | Value | |----------|----------------| | \( 3^{-2} \) | \( \frac{1}{9} \) | | \( 2^{-3} \) | \( \frac{1}{8} \) | | \( 10^{-1} \) | \( \frac{1}{10} \) | | \( x^{-2} \) | \( \frac{1}{x^2} \) | | \( x^{-3} \) | \( \frac{1}{x^3} \) | | \( x^{-4} \) | \( \frac{1}{x^4} \) | | \( x^{-13} \) | \( \frac{1}{x^{13}} \)| | \( x^{-1} \) | \( \frac{1}{x} \) |
If you need specific numerical values, you would need to provide a value for \( x \).