To factor the expression [tex]\( r^{27} - s^{30} \)[/tex], we can use the rule of factoring the difference of powers.
Given the expression [tex]\( r^{27} - s^{30} \)[/tex]:
1. Identify the greatest common powers that can be factored from the exponents. Since 27 and 30 share no common divisor greater than 1, we can express [tex]\( r^{27} \)[/tex] and [tex]\( s^{30} \)[/tex] in terms of lower powers as follows:
[tex]\[ r^{27} = (r^9)^3 \][/tex]
[tex]\[ s^{30} = (s^{10})^3 \][/tex]
2. Now, we see that [tex]\( r^{27} - s^{30} \)[/tex] can be written as [tex]\( (r^9)^3 - (s^{10})^3 \)[/tex].
3. Next, apply the difference of cubes formula, which states:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
In our case, let [tex]\( a = r^9 \)[/tex] and [tex]\( b = s^{10} \)[/tex].
4. Applying this formula, we get:
[tex]\[ (r^9)^3 - (s^{10})^3 = (r^9 - s^{10})((r^9)^2 + r^9 \cdot s^{10} + (s^{10})^2) \][/tex]
5. Simplify the terms inside the second set of parentheses:
[tex]\[ (r^9)^2 = r^{18} \][/tex]
[tex]\[ r^9 \cdot s^{10} = r^9 s^{10} \][/tex]
[tex]\[ (s^{10})^2 = s^{20} \][/tex]
6. Therefore, the expression simplifies to:
[tex]\[ r^{27} - s^{30} = (r^9 - s^{10})(r^{18} + r^9 s^{10} + s^{20}) \][/tex]
So, the correct factored form is:
[tex]\[
(r^9 - s^{10})(r^{18} + r^9 s^{10} + s^{20})
\][/tex]
Thus, the correct option is:
[tex]\[
\boxed{(r^9 - s^{10})(r^{18} + r^9 s^{10} + s^{20})}
\][/tex]
