Question

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Answer 1

$\mathfrak{\large{\underline{\underline{Answer:-}}}}$

$\boxed{\bf{(a + b)(a - b)( {a}^{2} - ab + {b}^{2})( {a}^{2} + ab + {b}^{2}) = {a}^{6} - {b}^{6}}}$

So, [b] is the answer.

$\mathfrak{\large{\underline{\underline{Explanation:-}}}}$

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

It can be written as :

$= ((a + b)( {a}^{2} - ab + {b}^{2}))((a - b)( {a}^{2} + ab + {b}^{2}))$

We know that, (x + y)(x² - xy + y²) = x³ + y³

Here, x = a, y = b

By substituting the values We have,

$=( {a}^{3} + {b}^{3})((a - b)( {a}^{2} + ab + {b}^{2}))$

We know that, (x - y)(x² + xy + y²) = x³ - y³

Here, x = a, y = b

By substituting the values We have,

$= ( {a}^{3} + {b}^{3})( {a}^{3} - {b}^{3})$

We know that, (x + y)(x - y) = x² - y²

Here x = a³ , y = b³

By substituting the values We have,

$= {( {a}^{3}) }^{2} - {( {b}^{3})}^{2}$

We know that $\tt{{ ({a}^{m} )}^{n} = {a}^{mn}}$

$= {a}^{3 \times 2} - {b}^{3 \times 2}$

$= {a}^{6} - {b}^{6}$

$\boxed{\bf{(a + b)(a - b)( {a}^{2} - ab + {b}^{2})( {a}^{2} + ab + {b}^{2}) = {a}^{6} - {b}^{6}}}$

So, [b] is the answer.

$\mathfrak{\large{\underline{\underline{Identities\:Used:-}}}}$

[1] (x + y)(x² - y + y²) = x³ + y³

[2](x - y)(x² + xy + y²) = x³ - y³

[3] (x + y)(x - y) = x² - y²

$\mathfrak{\large{\underline{\underline{Extra\: Information:-}}}}$

[1] (x + y)² = x² + y² + 2xy

[2] (x + y)(x - y) = x² - y²

[3] (x + a)(x + b) = x² + (a + b)x + ab

[4] (x + y)³ = x³ + y³ + 3xy(x + y)

[5] (x - y)³ x³ - y³ - 3xy(x - y)

[7] (x + y + z)² = x² + y² + z² + 2(xy + yz + xz)

[8] (x + y + z)(x² + y² + z² - xy - yz - yz) = x³ + y³ + z³ - 3xyz