`(r + 3s)^6` Use the Binomial Theorem to expand and simplify the expression.

We have to use the binomial theorem to expand the expression, so use the formula:

`(a+b)^n=sum_(k=0)^n((n),(k))a^(n-k)b^k`

`:.(r+3s)^6=((6),(0))r^(6-0)(3s)^0+((6),(1))r^(6-1)(3s)^1+((6),(2))r^(6-2)(3s)^2+((6),(3))r^(6-3)(3s)^3+((6),(4))r^(6-4)(3s)^4+((6),(5))r^(6-5)(3s)^5+((6),(6))r^(6-6)(3s)^6`

`=r^6+(6!)/(1!(6-1)!)r^5(3s)+(6!)/(2!(6-2)!)r^4(9s^2)+(6!)/(3!(6-3)!)r^3(27s^3)+(6!)/(4!(6-4)!)r^2(81s^4)+(6!)/(5!(6-5)!)r^1(243s^5)+729s^6`

`=r^6+(6*5!)/(5!)(3r^5s)+(6*5*4!)/(2*4!)(9r^4s^2)+(6*5*4*3!)/(3!*3*2*1)(27r^3s^3)+(6*5*4!)/(4!*2*1)(81r^2s^4)+(6*5!)/(5!)(243rs^5+729s^6`

`=r^6+18r^5s+135r^4s^2+540r^3s^3+1215r^2s^4+1458rs^5+729s^6`