To factor the expression [tex]\( r^{27} - s^{30} \)[/tex], we can use factoring techniques for differences of powers. Given that the powers involved are high, we can focus on patterns that appear in simpler cases and extend them accordingly.
The method to factor an expression of the form [tex]\(a^m - b^n\)[/tex] is to break it down into simpler polynomial factors. For our specific problem [tex]\( r^{27} - s^{30} \)[/tex], we aim to find such a factorization.
Here, we recognize that we are dealing with a difference of powers, and we can start by searching for a structure similar to differences of squares or cubes:
1. Notice that [tex]\(27\)[/tex] and [tex]\(30\)[/tex] have a common multiple with a factor that can simplify initial factorizations.
2. We rewrite [tex]\( r^{27} \)[/tex] and [tex]\( s^{30} \)[/tex] using their lower power bases:
[tex]\[
r^{27} - s^{30} = \left(r^9\right)^3 - \left(s^{10}\right)^3
\][/tex]
This transforms our problem into factoring a difference of cubes, which follows the formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
Here, let [tex]\( r^9 = A \)[/tex] and [tex]\( s^{10} = B \)[/tex]. Then our expression becomes:
[tex]\[ A^3 - B^3 \][/tex]
Now apply the difference of cubes formula:
[tex]\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \][/tex]
Substitute back [tex]\( A = r^9 \)[/tex] and [tex]\( B = s^{10} \)[/tex]:
[tex]\[ r^{27} - s^{30} = \left(r^9 - s^{10}\right)\left((r^9)^2 + r^9 \cdot s^{10} + (s^{10})^2\right) \][/tex]
[tex]\[ = (r^9 - s^{10})(r^{18} + r^9 \cdot s^{10} + s^{20}) \][/tex]
Therefore, the correct factored form of [tex]\( r^{27} - s^{30} \)[/tex] is:
[tex]\[
\boxed{\left(r^9 - s^{10}\right)\left(r^{18} + r^9 s^{10} + s^{20}\right)}
\][/tex]
Thus, the correct choice is:
[tex]\[
\left(r^9-s^{10}\right)\left(r^{18}+r^9 s^{10}+s^{20}\right)
\][/tex]
