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


{{{(a^-1 - b^-1)/(a^-3 - b^-3)}}}


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


{{{(1/a^1 - 1/b^1)/(1/a^3 - 1/b^3)}}} ----- {{{(1/a - 1/b)}}} ÷ {{{(1/a^3 - 1/b^3)}}} ---- {{{((b - a)/ab)}}} ÷ {{{((b^3 - a^3)/a^3b^3)}}} 


{{{((b - a)/ab) * (a^3b^3/(b^3 - a^3))}}} ----- Change division to multiplication and invert


{{{((b - a)/ab) * (a^3b^3/((b - a)(b^2 + ab + a^2)))}}} ---- Factoring {{{b^3 - a^3}}} 


{{{((cross(b - a))/cross(ab)) * (a^2cross(a^3)b^2cross(b^3)/((cross(b - a))(b^2 + ab + a^2)))}}} ------ {{{highlight_green((a^2b^2)/(b^2 + ab + a^2))}}}


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