1. Let's state the problem: simplify the expression $(3r-2s)(3r-2s)(3r-2s)(3r-2s)$. 2. Notice that this is the same factor $(3r-2s)$ multiplied by itself four times, so it is $(3r-2s)^4$. 3. To simplify $(3r-2s)^4$, it is easier to first find $(3r-2s)^2$ and then square the result. 4. Calculate $(3r-2s)^2 = (3r-2s)(3r-2s)$. Use the distributive property: $$ (3r)(3r) - (3r)(2s) - (2s)(3r) + (2s)(2s) = 9r^2 - 6rs - 6rs + 4s^2 $$ 5. Combine like terms: $$ 9r^2 - 12rs + 4s^2 $$ 6. Now, the original expression becomes $$ (3r-2s)^4 = ig((3r-2s)^2ig)^2 = (9r^2 - 12rs + 4s^2)^2 $$ 7. Next, expand $(9r^2 - 12rs + 4s^2)^2$ using the formula for the square of a trinomial $(a-b+c)^2 = a^2 - 2ab + 2ac + b^2 - 2bc + c^2$: $$ (9r^2)^2 - 2 imes 9r^2 imes 12rs + 2 imes 9r^2 imes 4s^2 + (-12rs)^2 - 2 imes (-12rs) imes 4s^2 + (4s^2)^2 $$ 8. Calculate each term: - $(9r^2)^2 = 81r^4$ - $-2 imes 9r^2 imes 12rs = -216r^3 s$ - $2 imes 9r^2 imes 4s^2 = 72r^2 s^2$ - $(-12rs)^2 = 144r^2 s^2$ - $-2 imes (-12rs) imes 4s^2 = 96r s^3$ - $(4s^2)^2 = 16s^4$ 9. Add all terms together: $$ 81r^4 - 216r^3 s + 72r^2 s^2 + 144r^2 s^2 + 96r s^3 + 16s^4 $$ 10. Combine like terms $72r^2 s^2 + 144r^2 s^2 = 216r^2 s^2$: $$ 81r^4 - 216r^3 s + 216r^2 s^2 + 96r s^3 + 16s^4 $$ This is the fully expanded form of $(3r-2s)^4$. Final answer: $$81r^4 - 216r^3 s + 216r^2 s^2 + 96r s^3 + 16s^4$$