Question

the product of (a+b)(a-b) ( {a }^{2} - ab + {b}^{2} )( {a }^{2} + ab + {b}^{2} )​

Answer 1

To find -

Find the product of -

( a + b )( a - b )( a² - ab + b² )( a² + ab + b² ) .

Solution -

( a + b )( a - b )( a² - ab + b² )( a² + ab + b² )

=> ( a² - b² )( a² - ab + b² )( a² + ab + b² )

=> ( a² - b² )[ ( a² + b² )² - a²b² ]

=> ( a² - b² ) [ a² + b⁴ + 2a²b² - a²b² ]

=> ( a² - b² )( a⁴ + a²b² + b⁴ )

=> a⁶ + a⁴b² + a²b⁴ - a⁴b² - a²b⁴ - b⁶

=> a⁶ - b⁶ .

This is the required answer.

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Additional Information -

( a + b )² = a² + 2ab + b²

( a - b )² = a² - 2ab + b²

( a + b )( a - b ) = a² - b²

( a + b )³ = a³ + 3ab ( a + b ) + b³

( a - b )³ = a³ - 3ab ( a + b ) - b³

( a + b + c )³ = a³ + b³ + c³ + 3 ( a + b )( b + c )( c + a )

a³ + b³ + c³ - 3abc = ( a + b + c )( a² + b² + c² - ab - bc - ca )

When a + b + c = 0 ,

a³ + b³ + c³ = 3abc .

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Answer 2

Given:-

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$\bf \implies(a+b)(a-b) ( {a }^{2} - ab + {b}^{2} )( {a }^{2} + ab + {b}^{2} )$

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To Find:-

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$\bf \implies \: The \: Solution$

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STEP BY STEP EXPLANATION:-

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$\bf \implies( {a}^{2} - {b}^{2} ) \times ( {a }^{2} - ab + {b}^{2} ) \times ( {a }^{2} + ab + {b}^{2} )$

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$\bf \implies( {a}^{2} - {b}^{2} ) \times (( {a }^{2} - ab + {b}^{2} ) - {a }^{2} \: {b}^{2} )$

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$\bf \implies( {a}^{2} - {b}^{2} ) \times ( {a}^{4} + 2 {a}^{2} \: {b}^{2} + {b}^{4} - {a}^{2} \: {b}^{2} )$

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$\bf \implies( {a}^{2} - {b}^{2} ) \times ( {a}^{4} + {a}^{2} \: {b}^{2} \: \: {b}^{4} )$

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$\bf \implies{a}^{6} - {b}^{6}$