How do you simplify #(sqrt(5k^4)+3sqrt(2k))/sqrt(3k^3)#?

1 Answer
Jul 19, 2017

Answer:

See a solution process belowL

Explanation:

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#