Question 926190
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{{{((a+b)^3-(a-b)^3)/((a+b)^2+(a-b)^2)}}}

Factor the numerator as the difference of two cubes.
Square out the terms in the denominator

{{{( (a+b)^""-(a-b) ) ( (a+b)^2+(a+b)(a-b)+(a-b)^2  )/(a^2+2ab+b^2+a^2-2ab+b^2)}}}

Continue simplifying:

{{{( a+b^""-a+b ) ( a^2+2ab+b^2+a^2-b^2+a^2-2ab+b^2)/(2a^2+2b^2)}}}

{{{2b(3a^2+b^2)/(2(a^2+b^2))}}}

{{{cross(2)b(3a^2+b^2)/(cross(2)(a^2+b^2))}}}

{{{b(3a^2+b^2)/(a^2+b^2)}}}

or

{{{b(3a^2+b^2)}}}{{{"÷"}}}{{{a^2+b^2}}}

Now since {{{3a^2 = b^2}}}

We can either give the final answer in terms of either a or b.

If we substitute {{{b^2}}} for {{{3a^2}}} and thus {{{b^2/3}}} for {{{a^2}}},

in

{{{b(3a^2+b^2)}}}{{{"÷"}}}{{{a^2+b^2}}}, we get:

{{{b(b^2+b^2)}}}{{{"÷"}}}{{{(b^2/3+b^2)}}}

{{{b(2b^2)}}}{{{"÷"}}}{{{(b^2/3+3b^2/3)}}}

{{{(2b^3)}}}{{{"÷"}}}{{{(4b^2/3)}}}

{{{(2b^3)}}}{{{"×"}}}{{{(3/(4b^2))}}}

{{{(6b^3/(4b^2))}}}

{{{3b/2}}}  <--- that is the answer in terms of b

Ifyou want the answer in terms of a, solve for b

{{{3a^2 = b^2}}}

{{{"" +- sqrt(3a^2)=b}}}

{{{"" +- a*sqrt(3)=b}}}

Then  {{{3b/2}}} becomes

{{{"" +- 3a*sqrt(3)/2}}}   <--- that is the answer in terms of a

Edwin</pre>