Question

Factorise : 216 a^3 - 2√2b^3 ​

Answer 1

$\large\underline{\sf{Solution-}}$

Given algebraic expression is

$\rm \: 216 {a}^{3} - 2 \sqrt{2} {b}^{3}$

can be rewritten as

$\rm \: = \: 6 \times 6 \times 6 \times {a}^{3} - \sqrt{2} \times \sqrt{2} \times \sqrt{2} \times {b}^{3}$

$\rm \: = \: {(6a)}^{3} - {( \sqrt{2}b) }^{3}$

We know,

$\boxed{ \rm{ \: {x}^{3} - {y}^{3} = (x - y)( {x}^{2} + xy + {y}^{2}) \: }}$

So, here

$\rm \: x = 6a$

$\rm \: y = \sqrt{2}b$

So, on substituting the values, we get

$\rm \: = \: (6a - \sqrt{2}b)[ {(6a)}^{2} + (6a)( \sqrt{2}b) + {( \sqrt{2} b)}^{2} ]$

$\rm \: = \: (6a - \sqrt{2}b)[ 36 {a}^{2} + 6 \sqrt{2}ab + {2b}^{2} ]$

Hence,

$\rm\implies \:216 {a}^{3} - 2 \sqrt{2} {b}^{3}= (6a - \sqrt{2}b)[ 36 {a}^{2} + 6 \sqrt{2}ab + {2b}^{2} ]$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$