First, we can rationalize the denominator by multiplying the expression by the appropriate form or #1# to remove the radical from the denominator while keeping the value of the expression the same:
#(sqrt(5k^4) + 3sqrt(2k))/sqrt(3k^3) => sqrt(3k^3)/sqrt(3k^3) xx (sqrt(5k^4) + 3sqrt(2k))/sqrt(3k^3) =>#
#(sqrt(3k^3)(sqrt(5k^4) + 3sqrt(2k)))/(sqrt(3k^3)sqrt(3k^3)) =>#
#(sqrt(3k^3)sqrt(5k^4) + 3sqrt(3k^3)sqrt(2k))/(sqrt(3k^3))^2 =>#
#(sqrt(3k^3)sqrt(5k^4) + 3sqrt(3k^3)sqrt(2k))/(3k^3)#
We can simplify the numerator using this rule for radicals:
#sqrt(color(red)(a)) * sqrt(color(blue)(b)) = sqrt(color(red)(a) * color(blue)(b))#
#(sqrt(3k^3)sqrt(5k^4) + 3sqrt(3k^3)sqrt(2k))/(3k^3) =>#
#(sqrt(3k^3 * 5k^4) + 3sqrt(3k^3 * 2k))/(3k^3) =>#
#(sqrt(15k^7) + 3sqrt(6k^4))/(3k^3)#
We can simplify the radicals using this rule for radicals:
#sqrt(color(red)(a) * color(blue)(b)) = sqrt(color(red)(a)) * sqrt(color(blue)(b))#
#(sqrt(15k^7) + 3sqrt(6k^4))/(3k^3) => (sqrt(k^6 * 15k) + 3sqrt(k^4 * 6))/(3k^3) =>#
#(k^3sqrt(15k) + 3k^2sqrt(6))/(3k^3)#
If necessary, we can simplify further as:
#(k^3sqrt(15k) + 3k^2sqrt(6))/(3k^3) => (k^3sqrt(15k))/(3k^3) + (3k^2sqrt(6))/(3k^3) =>#
#(color(red)(cancel(color(black)(k^3)))sqrt(15k))/(3color(red)(cancel(color(black)(k^3)))) + (color(red)(cancel(color(black)(3)))k^2sqrt(6))/(color(red)(cancel(color(black)(3)))k^3) => sqrt(15k)/3 + (k^2sqrt(6))/k^3 =>#
#sqrt(15k)/3 + (sqrt(6))/k#
