Given: #root(3)(k^d)=(root(3)(k))^5#

Cube both sides

#k^d=(root(3)(k))^15#

#color(brown)("Doing it the 'hard way'")#

Write #(root(3)(k))^15# as

#[ root(3)(k) xxroot(3)(k)xxroot(3)(k)] xx[ root(3)(k) xxroot(3)(k)xxroot(3)(k)] xx[ root(3)(k) xxroot(3)(k)xxroot(3)(k)] xx[ root(3)(k) xxroot(3)(k)xxroot(3)(k)]xx[ root(3)(k) xxroot(3)(k)xxroot(3)(k)]#

Which is the same as #k^5 #

Putting it all back together we have:

#k^d=k^5#

Then by direct comparison we have:

#d=5#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(brown)("Doing it the more straight forward way")#

Write #(root(3)(k))^15# as #k^((15/3)#

but #15-:3 = 5# giving

#(root(3)(k))^15# is the same as #k^5#

Then the rest is as above.