+1242 If a and b are nonzero real numbers such that \(\left| a \right| \ne \left| b \right|\), compute the value of the expression I\( \left( \frac{b^2}{a^2} + \frac{a^2}{b^2} - 2 \right) \times \left( \frac{a + b}{b - a} + \frac{b - a}{a + b} \right) \times \left( \frac{\frac{1}{a^2} + \frac{1}{b^2}}{\frac{1}{b^2} - \frac{1}{a^2}} - \frac{\frac{1}{b^2} - \frac{1}{a^2}}{\frac{1}{a^2} + \frac{1}{b^2}} \right). \)
+128474 First expression factored ( a/b - b/a)^2 = [ (a^2 - b^2)/ab]^2 = [a^2 - b^2]^2
__________
a^2b^2
Second expression
[ (a+ b)^2 + (b - a)^2 ] 2 [ a^2 + b^2 ]
__________________ = ______________
(b^2 - a^2) (b^2 - a^2 )
Third expression
[b^2 + a^2] [a^2 - b^2] [ a^2 + b^2]^2 - [a^2 - b^2] ^2
_________ - _________ = ________________________ =
[a^2 - b^2] [b^2 + a^2] [a^2 - b^2] [ b^2 + a^2]
[a^4 + 2a^2b^2 + b^4 - a^4 + 2a^2b^2 - b^4 ] 4a^2b^2
______________________________________ = ____________________
[a^2 - b^2 ] [ a^2 + b^2 ] [a^2 - b^2] [ a^2 + b^2]
Second expression x Third expression
2(a^2 + b^2 ) 4a^2b^2 8 a^2b^2
___________ x ____________________ = _____________________ (2)
(b^2 - a^2) (a^2 - b^2) (a^2 + b^2) (b^2 - a^2) ( a^2 - b^2)
(1) x (2) =
(a^2 - b^2)^2 8a^2b^2 8 (a^2 - b^2)^2
___________ x ___________________ = ____________________ =
a^2 b^2 (b^2 -a^2) (a^2 - b^2) - (a^2 - b^2)(a^2 - b^2)
-8 (a^2 - b^2 )^2
_____________ = -8
(a^2 - b^2) ^2
