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.
