Suppose that the greatest common factor of #a# and #b# is #k#
i.e. #(aVb)=k# using the notation in this question.
This means that
#color(white)("XXX")a=k * p#
and
#color(white)("XXX")b=k * q#
(for #k,p,q in NN)#
where
#color(white)("XXX")#the prime factors of #p#: #{p_1,p_2,...}#
#color(white)("XXX")#and
#color(white)("XXX")#the prime factors of #q#: #{q_1,q_2,...}#
#color(white)("XXXXXXXXXXXXXXXXXXXX")#have no common elements.
From the definition of #k# (above)
we have #(aVb)^n=k^n#
Further
#color(white)("XXX")a^n=(k * p)^n=k^n * p^n#
and
#color(white)("XXX")b^n=(k * q)^n=k^n * q^n#
where #p^n# and #q^n# can have no common prime factors (since #p# and #q# have no common prime factors.
Therefore
#color(white)("XXX")a^nVb^n=k^n#
...and
#(aVb)^n=a^nVb^n#
