Question

Find the product using suitable identity:
(Root x -1/root x)(root x +1/root x)(x+1/x)(x^2+1/x^2)(x^4+1/x^4)

Answer 1

Answer:

$\large\boxed{\sf{{x}^{8} - \frac{1}{ {x}^{8} } }}$

Step-by-step explanation:

To find the product of:

$( \sqrt{x} - \frac{1}{ \sqrt{x} } )( \sqrt{x} + \frac{1}{ \sqrt{x} } )(x + \frac{1}{x} )( {x}^{2} + \frac{1}{ {x}^{2} } )( {x}^{4} + \frac{1}{ {x}^{4} }) \\ \\ = ( {( \sqrt{x} )}^{2} - \frac{1}{ {( \sqrt{x} )}^{2} } )( x + \frac{1}{x} )( {x}^{2} + \frac{}{ {x}^{2} } )( {x}^{4} + \frac{1}{ {x}^{4} } ) \\ \\ = (x - \frac{1}{x} )(x + \frac{1}{x} )( {x}^{2} + \frac{1}{ {x}^{2} })( {x}^{4} + \frac{1}{ {x}^{4} } ) \\ \\ = ( {x}^{2} - \frac{1}{ {x}^{2} } )( {x}^{2} + \frac{1}{ {x}^{2} } )( {x}^{4} + \frac{1}{ {x}^{4} } ) \\ \\ = >( {( {x}^{2}) }^{2} - \frac{1}{ { ({x}^{2}) }^{2} } )( {x}^{4} + \frac{1}{ {x}^{4} } ) \\ \\ = ( {x}^{4} - \frac{1}{ {x}^{4} } )( {x}^{4} + \frac{1}{ {x}^{4} } ) \\ \\ = {( {x}^{4} )}^{2} - \frac{1}{ {( {x}^{4}) }^{2} } \\ \\ = {x}^{8} - \frac{1}{ {x}^{8} }$

Concept Map:-

• $(a+b)(a-b) = {a}^{2} - {b}^{2}$