How do you simplify #(-91a^4b+14ab )/ (-7ab)#?

Question

Genesis Allen · Accepted Answer

#(-91a^4b+14ab)/(-7ab)#

#= 13a^3-2#
#=(root(3)(13)a-root(3)(2))(root(3)(169)a^2+root(3)(26)a+root(3)(4))#

with exclusions #a!=0# and #b!=0# Both of the terms #-91a^4b# and #14ab# are divisible by #-7ab#, so we find: #(-91a^4b+14ab)/(-7ab) = 13a^3-2# with exclusions #a!=0# and #b!=0# Note that the exclusions are required because if #a=0# or #b=0# then the original expression is of the form #0/0# which is undefined. I guess that counts as 'simplified', but it can be factored too, using cube roots... For any Real numbers #x# and #y# note that: #root(3)(x) root(3)(y) = root(3)(xy)# We will use this below. The difference of cubes identity can be written: #A^3-B^3 = (A-B)(A^2+AB+B^2)# We can factor #13a^3-2# using this identity with #A=root(3)(13)a# and #b = root(3)(2)# as follows: #13a^3-2# #=(root(3)(13)a)^3-(root(3)(2))^3# #=(root(3)(13)a-root(3)(2))((root(3)(13)a)^2+(root(3)(13)a)(root(3)(2))+(root(3)(2))^2)# #=(root(3)(13)a-root(3)(2))(root(3)(169)a^2+root(3)(26)a+root(3)(4))#

Abigail Allen · Answer

undefined To simplify the expression (-91a^4b+14ab)/(-7ab), we can divide each term by -7ab. This results in: 13a^3 - 2.