SOLUTION: Hi This question has to do with negative exponents. I am trying to rewrite this expression without negative exponents. a^-1 - b^-1 ___________ a^-3 - b^-3 I believe the an

Algebra ->  Exponents -> SOLUTION: Hi This question has to do with negative exponents. I am trying to rewrite this expression without negative exponents. a^-1 - b^-1 ___________ a^-3 - b^-3 I believe the an      Log On


   

Question 639861: Hi
This question has to do with negative exponents. I am trying to rewrite this expression without negative exponents.
a^-1 - b^-1
___________
a^-3 - b^-3
I believe the answer is:
a^2b^2
_________________
a^2+ab+b^2
I just don't know the steps on how to get there.
Thank you
Regards
Mike
Found 3 solutions by DrBeeee, sachi, MathTherapy:
Answer by DrBeeee(684) About Me  (Show Source):
You can put this solution on YOUR website!
Your answer is correct. Here is how I get there.
The first operation we need to do is get rid of the negative exponents.
This is done by replacing them with their inverse (also called the reciprocal).
For example,
a^(-1) = 1/a
Likewise
b^(-1) = 1/b
and
a^(-3) = 1/a^3
and
b^(-3) = 1/b^3
Make these sustitutions in your expression yields
(1) (1/a - 1/b)/(1/a^3 - 1/b^3)
Now let's get rid of the fractions.
Multiply (1) by (a^3*b^3)/(a^3*b^3), which changes nothing because we are multiplying by one, and obtain
(2) (a^2*b^2)((b - a)/(b^3 - a^3))
The denominator of (2) factors into
(3) (b - a)*(b^2 + ab + a^2)
You can FOIL (3) to show that it is equal to the denominator of (2).
Substituting (3) into (2) which simplifies to
(4) (a^2*b^2)/(b^2 + ab + a^2)
The answer (4) is the same as your "belief", but now you know the rest of the story.
We can also go from (1) to (2) by cross multiplication to subtract the fractions in the numerator and denominator of (1).
The numerator of (1) is
(5) (1/a - 1/b) = (b-a)/(ab)
and the denominator of (1) is
(6) (1/a^3 - 1/b^3) = (b^3 - a^3)/(a^3*b^3)
Taking the ratio of (5)/(6) yields (2).




Answer by sachi(548) About Me  (Show Source):
You can put this solution on YOUR website!
a^-1 - b^-1
__________
a^-3 - b^-3
=(1/a-1/b)/(1/a^3-1/b^3)
=[(b-a)/ab]/[(b^3-a^3)/a^3b^3]
=[(b-a)/(b^3-a^3)]
=[(b-a)/(b-a)(b^2+a^2+ab)]
=[1/(b^2+a^2+ab)]
=a^2b^2/(b^2+a^2+ab) ans

Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!

Hi
This question has to do with negative exponents. I am trying to rewrite this expression without negative exponents.
a^-1 - b^-1
___________
a^-3 - b^-3
I believe the answer is:
a^2b^2
_________________
a^2+ab+b^2
I just don't know the steps on how to get there.
Thank you
Regards
Mike

%28a%5E-1+-+b%5E-1%29%2F%28a%5E-3+-+b%5E-3%29

Let's change these negative-exponent expressions to positive-exponent expressions:

%281%2Fa%5E1+-+1%2Fb%5E1%29%2F%281%2Fa%5E3+-+1%2Fb%5E3%29 ----- %281%2Fa+-+1%2Fb%29 ÷ %281%2Fa%5E3+-+1%2Fb%5E3%29 ---- %28%28b+-+a%29%2Fab%29 ÷ %28%28b%5E3+-+a%5E3%29%2Fa%5E3b%5E3%29

%28%28b+-+a%29%2Fab%29+%2A+%28a%5E3b%5E3%2F%28b%5E3+-+a%5E3%29%29 ----- Change division to multiplication and invert

%28%28b+-+a%29%2Fab%29+%2A+%28a%5E3b%5E3%2F%28%28b+-+a%29%28b%5E2+%2B+ab+%2B+a%5E2%29%29%29 ---- Factoring b%5E3+-+a%5E3

------ highlight_green%28%28a%5E2b%5E2%29%2F%28b%5E2+%2B+ab+%2B+a%5E2%29%29

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