Question

find the product of using identity (a-1/2)(a+1/a)(a^2+1/a^2)(a^4+1/a^4)

Answer 1

$\textbf{\huge{\underline{ Hello Mate! }}}$

We will use identity $x^2 - y^2$ = $\textsf{\blue{ ( x + y )( x - y ) }}$

$= (a - \frac{1}{a} )(a + \frac{1}{a} )( {a}^{2} + \frac{1}{ {a}^{2} } )( {a}^{4} + \frac{1}{ {a}^{4} } ) \\ \\= ( {a}^{2} - \frac{1}{ {a}^{2} } )( {a}^{2} + \frac{1}{ {a}^{2} } )( {a}^{4} + \frac{1}{ {a}^{4} } ) \\ \\= ( {a}^{4} - \frac{1}{ {a}^{4} } )( {a}^{4} + \frac{1}{ {a}^{4} } ) \\ \\ = {a}^{8} - \frac{1}{ {a}^{8} }$

$\textsf{\green{ What I did? }}$

When,

$= ({x}^{2} - {y}^{2} )( {x}^{2} + {y}^{2} ) \\ = { ({x}^{2} )}^{2} - { ({y}^{2}) }^{2} \\ = {x}^{4} - {y}^{4}$

$\textsf{\red{ Hope It helps }}$

$\textbf{ Have great future ahead! }$

Answer 2

here is your answer OK dude Given


using identity:
(x - y) (x + y) = x2 - y2

(a - 1/a)(a + 1/a) (a2 + 1/a2)(a4 + 1/a4)
= [(a)2 - (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
= a8 - 1/a8