Solution:

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We know the algebraic identity:

(x+y)(x-y) = x² - y²

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Here ,

(a+1/a)(a-1/a)(a²+1/a²)(a⁴+1/a⁴)

=[(a+1/a)(a-1/a)](a²+1/a²)(a⁴+1/a⁴)

=[a²-(1/a)²](a²+1/a²)(a⁴+1/a⁴)

=[(a²-1/a²)(a²+1/a²)](a⁴+1/a⁴)

= [(a²)²-(1/a²)²](a⁴+1/a⁴)

= (a⁴-1/a⁴)(a⁴+1/a⁴)

= (a⁴)²-(1/a⁴)²

=

Therefore,

(a+1/a)(a-1/a)(a²+1/a²)(a⁴+1/a⁴) =

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