To find the simplified form of the expression [tex]\( r^{-7} + s^{-12} \)[/tex], let's go through the problem step-by-step.
1. Understand Negative Exponents:
- For any non-zero number [tex]\( a \)[/tex] and a positive integer [tex]\( n \)[/tex], [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex].
2. Apply Negative Exponent Rules:
- For [tex]\( r^{-7} \)[/tex], it can be rewritten as [tex]\( \frac{1}{r^7} \)[/tex].
- Similarly, for [tex]\( s^{-12} \)[/tex], it can be rewritten as [tex]\( \frac{1}{s^{12}} \)[/tex].
3. Add the Simplified Terms:
- Combining these two expressions results in:
[tex]\[
r^{-7} + s^{-12} = \frac{1}{r^7} + \frac{1}{s^{12}}
\][/tex]
So, the simplified form of [tex]\( r^{-7} + s^{-12} \)[/tex] is:
[tex]\[
\boxed{\frac{1}{r^7} + \frac{1}{s^{12}}}
\][/tex]
Given the options:
- [tex]\(\frac{1}{r^7 s^{12}}\)[/tex]
- [tex]\(-r^7 - s^{12}\)[/tex]
- [tex]\(\frac{r^7}{s^{12}}\)[/tex]
- [tex]\(\frac{1}{r^7} + \frac{1}{s^{12}}\)[/tex]
The correct simplified form is:
[tex]\[
\boxed{\frac{1}{r^7} + \frac{1}{s^{12}}}
\][/tex]
