we conclude that:
![\left(5^{-2}\right)^2=5^2\cdot \:\:\:5^{-6}=\frac{1}{5}\cdot \:\:\frac{1}{5}\cdot \:\:\frac{1}{5}\cdot \:\:\frac{1}{5}](/tpl/images/1023/9989/56ef1.png)
Hence, options A and D are true.
Step-by-step explanation:
Given the expression
![\frac{1}{5}\cdot \frac{1}{5}\cdot \frac{1}{5}\cdot \frac{1}{5}](/tpl/images/1023/9989/3cbe7.png)
![\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{a}{b}\cdot \frac{c}{d}=\frac{a\:\cdot \:c}{b\:\cdot \:d}](/tpl/images/1023/9989/c6008.png)
![\frac{1}{5}\cdot \frac{1}{5}\cdot \frac{1}{5}\cdot \frac{1}{5}=\frac{1\cdot \:\:1\cdot \:\:1\cdot \:\:1}{5\cdot \:\:5\cdot \:\:5\cdot \:\:5}](/tpl/images/1023/9989/ce336.png)
![=\frac{1}{5\cdot \:5\cdot \:5\cdot \:5}](/tpl/images/1023/9989/0bdf7.png)
![=\frac{1}{5^4}](/tpl/images/1023/9989/ee367.png)
![=\frac{1}{625}](/tpl/images/1023/9989/5a3ee.png)
Checking options B and C:
Option B
Option C
![\frac{5^1}{5^4}=\frac{5}{625}=\frac{1}{125}](/tpl/images/1023/9989/8c252.png)
Thus, options B and C are not equivalent!
Checking Option A
Option A
![\left(5^{-2}\right)^2](/tpl/images/1023/9989/b3b8d.png)
![\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc},\:\quad \:a\ge 0](/tpl/images/1023/9989/1894c.png)
![\left(5^{-2}\right)^2=5^{-2\cdot \:2}](/tpl/images/1023/9989/1deb4.png)
![=5^{-4}](/tpl/images/1023/9989/f618d.png)
![\mathrm{Apply\:exponent\:rule}:\quad \:a^{-b}=\frac{1}{a^b}](/tpl/images/1023/9989/c8c07.png)
![=\frac{1}{5^4}](/tpl/images/1023/9989/ee367.png)
![=\frac{1}{625}](/tpl/images/1023/9989/5a3ee.png)
Now, checking option D
Option D
![5^2\cdot \:5^{-6}](/tpl/images/1023/9989/f7d46.png)
![\mathrm{Apply\:exponent\:rule}:\quad \:a^b\cdot \:a^c=a^{b+c}](/tpl/images/1023/9989/aa1f5.png)
![5^2\cdot 5^{-6}=5^{2-6}](/tpl/images/1023/9989/02242.png)
![=5^{-4}](/tpl/images/1023/9989/f618d.png)
![\mathrm{Apply\:exponent\:rule}:\quad \:a^{-b}=\frac{1}{a^b}](/tpl/images/1023/9989/c8c07.png)
![=\frac{1}{5^4}](/tpl/images/1023/9989/ee367.png)
![=\frac{1}{625}](/tpl/images/1023/9989/5a3ee.png)
Therefore, we conclude that:
![\left(5^{-2}\right)^2=5^2\cdot \:\:\:5^{-6}=\frac{1}{5}\cdot \:\:\frac{1}{5}\cdot \:\:\frac{1}{5}\cdot \:\:\frac{1}{5}](/tpl/images/1023/9989/56ef1.png)
Hence, options A and D are true.