Which is the simplified form of [tex]$r^{-7} + s^{-12}$[/tex]?

A. [tex]\frac{1}{r^7 s^{12}}[/tex]
B. [tex]-r^7 - s^{12}[/tex]
C. [tex]\frac{r^7}{s^{12}}[/tex]
D. [tex]\frac{1}{r^7} + \frac{1}{s^{12}}[/tex]



Answer :

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]