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

Answer 1

#(-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))#
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Answer 2

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

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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