To factor the expression [tex]\( r^{27} - s^{30} \)[/tex], one can use the properties of polynomials and factorization.
Let's start by observing that both terms [tex]\( r^{27} \)[/tex] and [tex]\( s^{30} \)[/tex] have powers that are multiples of 3. We can then factor this as a difference of cubes. Specifically, we note that:
[tex]\[ r^{27} = (r^9)^3 \][/tex]
[tex]\[ s^{30} = (s^{10})^3 \][/tex]
By the difference of cubes formula, [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex], we identify [tex]\( a = r^9 \)[/tex] and [tex]\( b = s^{10} \)[/tex].
Applying the formula:
[tex]\[ r^{27} - s^{30} = (r^9)^3 - (s^{10})^3 \][/tex]
[tex]\[ = (r^9 - s^{10})((r^9)^2 + r^9(s^{10}) + (s^{10})^2) \][/tex]
Simplifying inside the second bracket:
[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]
Hence, we get:
[tex]\[ r^{27} - s^{30} = (r^9 - s^{10})(r^{18} + r^9 s^{10} + s^{20}) \][/tex]
Thus, the factored form of [tex]\( r^{27} - s^{30} \)[/tex] is:
[tex]\[ (r^9 - s^{10})(r^{18} + r^9 s^{10} + s^{20}) \][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\left(r^9 - s^{10}\right)\left(r^{18} + r^9 s^{10} + s^{20}\right)}
\][/tex]
