Which expressions can be simplified as [tex]\frac{625}{n^{12}}[/tex]? Check all that apply.

A. [tex]\left(5 n^{-3}\right)^4[/tex]

B. [tex]\left(5 n^{-3}\right)^{-4}[/tex]

C. [tex]\left(5 n^{-4}\right)^3[/tex]

D. [tex]\left(25 n^{-6}\right)^{-2}[/tex]

E. [tex]\left(25 n^{-6}\right)^2[/tex]



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.