I can't figure out what am I doing wrong ..

Question

Please help me figuring out what have i done wrong.

(a/b - b/a)^2  *  (ab/b-a)^2

Here are my steps:

a^2 * b^2 = (a*b)^2

(a/b - b/a * ab/b-a)^2   

=(a-b)(b-a)(ab) / b-a

=(a-b)(-a+b)(ab) / b-a

=(-a+b)^2 (ab) /b-a

=(-a^2-ab+ba+b^2)^2 / b-a=

=( - a^2 + b^2)^2 / b-a

=(b-a)^2 = (-a+b)^2

The final result in my textbook is (a+b)^2. I can't find what am I doing wrong, please help.

Answer 1

First a/b - b/a=(a^2-b^2)/ab. Now square it: (a^2-b^2)^2/(ab)^2.

Now multiply by (ab)^2/(b-a)^2. The (ab)^2 cancel out leaving (a^2-b^2)^2/(b-a)^2.

We can factorise the top: ((a-b)(a+b))^2=(a-b)^2 * (a+b)^2/(b-a)^2.

(b-a)^2 is the same as (a-b)^2, so again we can cancel out (a-b)^2 leaving (a+b)^2.