Answer: OPTION D.
Step-by-step explanation:
The missing options are:
[tex]A. 5^3*5^{-5}\\B. 5^{-1}*5^{-1}\\C. 5^{-3}*5\\ D. 5^{-2}*5^4[/tex]
In order to solve this exercise it is necessary to remember the following:
1. The Product of powers property states that:
[tex](a^m)(a^n)=a^{(n+m)}[/tex]
2. The Quotient of powers property states that:
[tex]\frac{a^m}{a^n}=a^{(m-n)}[/tex]
3. The Negative exponent rule states the following:
[tex]a^{-n}=\frac{1}{a^n}[/tex]
Then, knowing those properties you can check each option by applying them, in order to find which expression given in the exercise is equal to the fraction [tex]\frac{1}{25}[/tex].
You get:
[tex]A.\ 5^3*5^{-5}=5^{(3-5)}=5^{-2}=\frac{1}{5^2}=\frac{1}{25} \\\\B.\ 5^{-1}*5^{-1}=5^{(-1-1)}=5^{-2}=\frac{1}{5^2}=\frac{1}{25}\\\\C.\ 5^{-3}*5=5^{(-3+1)}=5^{-2}=\frac{1}{5^2}=\frac{1}{25}\\\\ D.\ 5^{-2}*5^4=5^{(-2+4)}=5^2=25[/tex]
Therefore, you can identify that the expression given in the OPTION D is not equal to [tex]\frac{1}{25}[/tex].
