Question
\[ \text { Fully simplify }\left(4 r^{-6}\right)^{-2} \] Give your answer without any negative indices.
Ask by Sanders Ortiz. in Spain
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\(\frac{r^{12}}{16}\)
Solution
To fully simplify \(\left(4 r^{-6}\right)^{-2}\) without any negative exponents, follow these steps:
1. **Apply the power to both the coefficient and the variable:**
\[
\left(4 r^{-6}\right)^{-2} = 4^{-2} \cdot \left(r^{-6}\right)^{-2}
\]
2. **Simplify each part separately:**
- \(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\)
- \(\left(r^{-6}\right)^{-2} = r^{(-6) \times (-2)} = r^{12}\)
3. **Combine the simplified parts:**
\[
\frac{1}{16} \cdot r^{12} = \frac{r^{12}}{16}
\]
So, the fully simplified expression without any negative exponents is:
\[
\frac{r^{12}}{16}
\]
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Simplify this solution Bonus Knowledge
To simplify \(\left(4 r^{-6}\right)^{-2}\), we first apply the exponent to both the coefficient and the variable: \[ \left(4 r^{-6}\right)^{-2} = 4^{-2} (r^{-6})^{-2} \] Calculating \(4^{-2}\): \[ 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \] Next, calculating \((r^{-6})^{-2}\): \[ (r^{-6})^{-2} = r^{12} \] Now, putting everything together, we have: \[ \left(4 r^{-6}\right)^{-2} = \frac{1}{16} r^{12} \] To express it without negative indices, we can write the result as: \[ \frac{r^{12}}{16} \] Thus, the fully simplified expression is: \[ \frac{r^{12}}{16} \]
