Answer:

[tex]64 {m}^{3} - 343 {n}^{3} [/tex]

[tex] {(4m)}^{3} - {(7n)}^{3} [/tex]

[tex] {a}^{3} - {b}^{3} = (a - b)( {a}^{2} + ab + {b}^{2} )[/tex]

a = 4m, b = 7n

[tex] = (4m - 7n)( {(4m)}^{2} + (4m \times 7n) + {(7n)}^{2} )[/tex]

[tex] = (4m - 7n)(16 {m}^{2} + 28mn + 49 {n}^{2} )[/tex]

Verified answer

Answer:

[tex]\boxed{ \bf\: {64m}^{3} - {343n}^{3} = (4m - 7n)\left({16m}^{2} + 28mn + {49n}^{2}\right) \: }\\ [/tex]

Step-by-step explanation:

Given algebraic expression is

[tex]\rm \: {64m}^{3} - {343n}^{3} \\ [/tex]

[tex]\rm \: = \: {(4m)}^{3} - {(7n)}^{3} \\ [/tex]

We know,

[tex]\boxed{ \sf\: {x}^{3} - {y}^{3} = (x - y)( {x}^{2} + xy + {y}^{2} ) \: } \\ [/tex]

So, using this algebraic identity, we get

[tex]\rm \: = \: (4m - 7n)[ {(4m)}^{2} + (4m)(7n) + {(7n)}^{2}] \\ [/tex]

[tex]\rm \: = \: (4m - 7n)({16m}^{2} + 28mn + {49n}^{2}) \\ [/tex]

Hence,

[tex]\implies\boxed{ \bf\: {64m}^{3} - {343n}^{3} = (4m - 7n)\left({16m}^{2} + 28mn + {49n}^{2}\right) \: }\\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Additional Information

[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{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}[/tex]