6. If \( n^{-4}=\frac{1}{28,561} \), what could be the value of \( n \) ? (A) \( \frac{1}{169} \) (B) \( -\frac{1}{13} \) (C) \( \frac{1}{28,561} \) (D) \( \frac{1}{13} \) (E) 13

To solve the equation \( n^{-4} = \frac{1}{28,561} \), we can follow these steps: 1. **Rewrite the equation**: The expression \( n^{-4} \) can be rewritten as \( \frac{1}{n^4} \). Therefore, we have: \[ \frac{1}{n^4} = \frac{1}{28,561} \] 2. **Cross-multiply**: This gives us: \[ n^4 = 28,561 \] 3. **Take the fourth root**: To find \( n \), we need to take the fourth root of both sides: \[ n = \sqrt[4]{28,561} \] Next, we will calculate \( \sqrt[4]{28,561} \) to find the possible values of \( n \). Let's calculate \( \sqrt[4]{28,561} \). First, we can find \( \sqrt{28,561} \) and then take the square root of that result. Calculating \( \sqrt{28,561} \): \[ \sqrt{28,561} = 169 \] Now, we take the square root of \( 169 \): \[ \sqrt{169} = 13 \] Thus, we have: \[ n = \sqrt[4]{28,561} = 13 \] Now, let's check the options provided: - (A) \( \frac{1}{169} \) - (B) \( -\frac{1}{13} \) - (C) \( \frac{1}{28,561} \) - (D) \( \frac{1}{13} \) - (E) 13 The value of \( n \) that we found is \( 13 \), which corresponds to option (E). Therefore, the answer is: \[ \boxed{13} \]