First, we can rationalize the denominator by multiplying the expression by the appropriate form or 11 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) =>√5k4+3√2k√3k3⇒√3k3√3k3×√5k4+3√2k√3k3⇒
(sqrt(3k^3)(sqrt(5k^4) + 3sqrt(2k)))/(sqrt(3k^3)sqrt(3k^3)) =>√3k3(√5k4+3√2k)√3k3√3k3⇒
(sqrt(3k^3)sqrt(5k^4) + 3sqrt(3k^3)sqrt(2k))/(sqrt(3k^3))^2 =>√3k3√5k4+3√3k3√2k(√3k3)2⇒
(sqrt(3k^3)sqrt(5k^4) + 3sqrt(3k^3)sqrt(2k))/(3k^3)√3k3√5k4+3√3k3√2k3k3
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))√a⋅√b=√a⋅b
(sqrt(3k^3)sqrt(5k^4) + 3sqrt(3k^3)sqrt(2k))/(3k^3) =>√3k3√5k4+3√3k3√2k3k3⇒
(sqrt(3k^3 * 5k^4) + 3sqrt(3k^3 * 2k))/(3k^3) =>√3k3⋅5k4+3√3k3⋅2k3k3⇒
(sqrt(15k^7) + 3sqrt(6k^4))/(3k^3)√15k7+3√6k43k3
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))√a⋅b=√a⋅√b
(sqrt(15k^7) + 3sqrt(6k^4))/(3k^3) => (sqrt(k^6 * 15k) + 3sqrt(k^4 * 6))/(3k^3) =>√15k7+3√6k43k3⇒√k6⋅15k+3√k4⋅63k3⇒
(k^3sqrt(15k) + 3k^2sqrt(6))/(3k^3)k3√15k+3k2√63k3
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) =>k3√15k+3k2√63k3⇒k3√15k3k3+3k2√63k3⇒
(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
