Question

Find the product using suitable identity. (x-1/x)(x+1/x)(x2+1/x2)(x4+1/x4)

Answer 1

Here is the answer hope so it will be helpful

Answer 2

$\displaystyle \sf{ \bigg( {x}^{} - \frac{1}{ {x}^{} } \bigg)\bigg( {x}^{} + \frac{1}{ {x}^{} } \bigg)\bigg( {x}^{2} + \frac{1}{ {x}^{2} } \bigg)\bigg( {x}^{4} + \frac{1}{ {x}^{4} } \bigg) } = {x}^{8} - \frac{1}{ {x}^{8} }$

Given :

$\displaystyle \sf{ \bigg( {x}^{} - \frac{1}{ {x}^{} } \bigg)\bigg( {x}^{} + \frac{1}{ {x}^{} } \bigg)\bigg( {x}^{2} + \frac{1}{ {x}^{2} } \bigg)\bigg( {x}^{4} + \frac{1}{ {x}^{4} } \bigg) }$

To find :

Simplify the given expression

Formula :

a² - b² = ( a + b ) ( a - b )

Solution :

Step 1 of 2 :

Write down the given expression

The given expression is

$\displaystyle \sf{ \bigg( {x}^{} - \frac{1}{ {x}^{} } \bigg)\bigg( {x}^{} + \frac{1}{ {x}^{} } \bigg)\bigg( {x}^{2} + \frac{1}{ {x}^{2} } \bigg)\bigg( {x}^{4} + \frac{1}{ {x}^{4} } \bigg) }$

Step 2 of 2 :

Simplify the expression

$\displaystyle \sf{ \bigg( {x}^{} - \frac{1}{ {x}^{} } \bigg)\bigg( {x}^{} + \frac{1}{ {x}^{} } \bigg)\bigg( {x}^{2} + \frac{1}{ {x}^{2} } \bigg)\bigg( {x}^{4} + \frac{1}{ {x}^{4} } \bigg) }$

$\displaystyle \sf{ = \bigg \{ {(x)}^{2} - {\bigg( \frac{1}{ {x}^{} } \bigg) }^{2} \bigg \} \bigg( {x}^{2} + \frac{1}{ {x}^{2} } \bigg)\bigg( {x}^{4} + \frac{1}{ {x}^{4} } \bigg) }$

$\displaystyle \sf{ = \bigg( {x}^{2} - \frac{1}{ {x}^{2} } \bigg) \bigg( {x}^{2} + \frac{1}{ {x}^{2} } \bigg)\bigg( {x}^{4} + \frac{1}{ {x}^{4} } \bigg) }$

$\displaystyle \sf{ = \bigg \{ {( {x}^{2} )}^{2} - {\bigg( \frac{1}{ {x}^{2} } \bigg) }^{2} \bigg \} \bigg( {x}^{4} + \frac{1}{ {x}^{4} } \bigg) }$

$\displaystyle \sf{ = \bigg( {x}^{4} - \frac{1}{ {x}^{4} } \bigg) \bigg( {x}^{4} + \frac{1}{ {x}^{4} } \bigg) }$

$\displaystyle \sf{ = \bigg \{ {( {x}^{4} )}^{2} - {\bigg( \frac{1}{ {x}^{4} } \bigg) }^{2} \bigg \} }$

$\displaystyle \sf{ = {x}^{8} - \frac{1}{ {x}^{8} } }$

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