Answer :
Let's analyze each expression step by step to see if it simplifies to [tex]\(\frac{625}{n^{12}}\)[/tex]:
1. [tex]\(\left(5 n^{-3}\right)^4\)[/tex]
We can apply the power of a power rule to simplify this:
[tex]\[ \left(5 n^{-3}\right)^4 = 5^4 \cdot \left(n^{-3}\right)^4 = 625 \cdot n^{-12} = \frac{625}{n^{12}} \][/tex]
Conclusion: This expression can be simplified as [tex]\(\frac{625}{n^{12}}\)[/tex].
2. [tex]\(\left(5 n^{-3}\right)^{-4}\)[/tex]
We can apply the power of a power rule to simplify this:
[tex]\[ \left(5 n^{-3}\right)^{-4} = 5^{-4} \cdot \left(n^{-3}\right)^{-4} = \frac{1}{5^4} \cdot n^{12} = \frac{1}{625} \cdot n^{12} = \frac{n^{12}}{625} \][/tex]
Conclusion: This expression cannot be simplified as [tex]\(\frac{625}{n^{12}}\)[/tex].
3. [tex]\(\left(5 n^{-4}\right)^3\)[/tex]
We can apply the power of a power rule to simplify this:
[tex]\[ \left(5 n^{-4}\right)^3 = 5^3 \cdot \left(n^{-4}\right)^3 = 125 \cdot n^{-12} = \frac{125}{n^{12}} \][/tex]
Conclusion: This expression cannot be simplified as [tex]\(\frac{625}{n^{12}}\)[/tex].
4. [tex]\(\left(25 n^{-6}\right)^{-2}\)[/tex]
We can apply the power of a power rule to simplify this:
[tex]\[ \left(25 n^{-6}\right)^{-2} = 25^{-2} \cdot \left(n^{-6}\right)^{-2} = \frac{1}{25^2} \cdot n^{12} = \frac{1}{625} \cdot n^{12} = \frac{n^{12}}{625} \][/tex]
Conclusion: This expression cannot be simplified as [tex]\(\frac{625}{n^{12}}\)[/tex].
5. [tex]\(\left(25 n^{-6}\right)^2\)[/tex]
We can apply the power of a power rule to simplify this:
[tex]\[ \left(25 n^{-6}\right)^2 = 25^2 \cdot \left(n^{-6}\right)^2 = 625 \cdot n^{-12} = \frac{625}{n^{12}} \][/tex]
Conclusion: This expression can be simplified as [tex]\(\frac{625}{n^{12}}\)[/tex].
### Summary
The expressions that can be simplified to [tex]\(\frac{625}{n^{12}}\)[/tex] are:
- [tex]\(\left(5 n^{-3}\right)^4\)[/tex]
- [tex]\(\left(25 n^{-6}\right)^2\)[/tex]
Thus, the indices of the expressions which can be simplified to [tex]\(\frac{625}{n^{12}}\)[/tex] are expressions 1 and 5.
1. [tex]\(\left(5 n^{-3}\right)^4\)[/tex]
We can apply the power of a power rule to simplify this:
[tex]\[ \left(5 n^{-3}\right)^4 = 5^4 \cdot \left(n^{-3}\right)^4 = 625 \cdot n^{-12} = \frac{625}{n^{12}} \][/tex]
Conclusion: This expression can be simplified as [tex]\(\frac{625}{n^{12}}\)[/tex].
2. [tex]\(\left(5 n^{-3}\right)^{-4}\)[/tex]
We can apply the power of a power rule to simplify this:
[tex]\[ \left(5 n^{-3}\right)^{-4} = 5^{-4} \cdot \left(n^{-3}\right)^{-4} = \frac{1}{5^4} \cdot n^{12} = \frac{1}{625} \cdot n^{12} = \frac{n^{12}}{625} \][/tex]
Conclusion: This expression cannot be simplified as [tex]\(\frac{625}{n^{12}}\)[/tex].
3. [tex]\(\left(5 n^{-4}\right)^3\)[/tex]
We can apply the power of a power rule to simplify this:
[tex]\[ \left(5 n^{-4}\right)^3 = 5^3 \cdot \left(n^{-4}\right)^3 = 125 \cdot n^{-12} = \frac{125}{n^{12}} \][/tex]
Conclusion: This expression cannot be simplified as [tex]\(\frac{625}{n^{12}}\)[/tex].
4. [tex]\(\left(25 n^{-6}\right)^{-2}\)[/tex]
We can apply the power of a power rule to simplify this:
[tex]\[ \left(25 n^{-6}\right)^{-2} = 25^{-2} \cdot \left(n^{-6}\right)^{-2} = \frac{1}{25^2} \cdot n^{12} = \frac{1}{625} \cdot n^{12} = \frac{n^{12}}{625} \][/tex]
Conclusion: This expression cannot be simplified as [tex]\(\frac{625}{n^{12}}\)[/tex].
5. [tex]\(\left(25 n^{-6}\right)^2\)[/tex]
We can apply the power of a power rule to simplify this:
[tex]\[ \left(25 n^{-6}\right)^2 = 25^2 \cdot \left(n^{-6}\right)^2 = 625 \cdot n^{-12} = \frac{625}{n^{12}} \][/tex]
Conclusion: This expression can be simplified as [tex]\(\frac{625}{n^{12}}\)[/tex].
### Summary
The expressions that can be simplified to [tex]\(\frac{625}{n^{12}}\)[/tex] are:
- [tex]\(\left(5 n^{-3}\right)^4\)[/tex]
- [tex]\(\left(25 n^{-6}\right)^2\)[/tex]
Thus, the indices of the expressions which can be simplified to [tex]\(\frac{625}{n^{12}}\)[/tex] are expressions 1 and 5.
