Question

solve using identity​

Answer 1

$\footnotesize \bf{1 \cancel{ .} Simplify \: the \: Following}$

• $\small\sf \left( \dfrac{x}{2} + \dfrac{y}{3} \right)^{3} - \left( \dfrac{x}{2} - \dfrac{y}{3} \right)^{3}$

$\footnotesize{ \sf By \: Using \: Identity \: } \bf _{ \mapsto \: \: (a - b) ^{3} = {a}^{3} - {b}^{3} - 3 {a}^{2} b+ 3 a{b}^{2} }$

${ \: \: \: \sf \: \: : \implies \: { \left( \dfrac{x}{2} \right)}^{3} + {\left( \dfrac{y}{3}\right)}^{3} + 3 \times {\left(\dfrac{x}{2}\right)}^{2} \times \dfrac{y}{3} + 3 \times\dfrac{x}{2} \times {\left(\dfrac{y}{3}\right)}^{2} - \bigg\{{ \left( \dfrac{x}{2} \right)}^{3} - {\left( \dfrac{y}{3}\right)}^{3} - 3 \times {\left(\dfrac{x}{2}\right)}^{2} \times \dfrac{y}{3} + 3 \times\dfrac{x}{2} \times {\left(\dfrac{y}{3}\right)}^{2} \bigg\} }$

${{ \: \: \: \sf \: \: : \implies \: { \left( \dfrac{x}{2} \right)}^{3} + { \dfrac{ {y}^{3} }{27}} + \dfrac{3 {x}^{2}y }{ {2}^{2} \times 3 } + \dfrac{3x {y}^{2} }{2 \times { 3}^{2} } - \bigg\{ \left( \dfrac{x}{2} \right)}^{3} \sf - { \dfrac{ {y}^{3} }{27}} - \dfrac{3 {x}^{2}y }{ {2}^{2} \times 3 } + \dfrac{3x {y}^{2} }{2 \times { 3}^{2}}\bigg\} }$

${{ \: \: \: \sf \: \: : \implies \: { \cancel{\left( \dfrac{x}{2} \right)}^{3} } + { \dfrac{ {y}^{3} }{27}} + \dfrac{3 {x}^{2}y }{ {2}^{2} \times 3 } + \cancel{ \dfrac{3x {y}^{2} } {2 \times { 3}^{2}}} - \cancel{\left( \dfrac{x}{2} \right)}}^{3} \sf + { \dfrac{ {y}^{3} }{27}} + \dfrac{3 {x}^{2}y }{ {2}^{2} \times 3 } - \cancel{ \dfrac{3x {y}^{2} }{2 \times { 3}^{2}} }}$

${{ \: \: \: \sf \: \: : \implies \: { \dfrac{ {y}^{3} }{27}} + \dfrac{3 {x}^{2}y }{ 12 }}} \sf + { \dfrac{ {y}^{3} }{27}} + \dfrac{3 {x}^{2}y }{ 12 }$

${{ \: \: \: \sf \: \: : \implies \: { \dfrac{ 2{y}^{3} }{27}} + \dfrac{6 {x}^{2}y }{ 12 }}}$

${{ \: \: \: \sf \: \: : \implies \: { \dfrac{ 2{y}^{3} }{27}} + \dfrac{ {x}^{2}y }{ 2 }}}$

${\tt \left( \dfrac{x}{2} + \dfrac{y}{3} \right)^{3} - \left( \dfrac{x}{2} - \dfrac{y}{3} \right)^{3} = \bf { \dfrac{ 2{y}^{3} }{27}} + \dfrac{ {x}^{2}y }{ 2 }}$

• $\begin{gathered}\blue{\begin{gathered}\tiny\begin{gathered}\small{\small{\small{\small{\small{\small{\small{\small{\small{\small{\begin{gathered}\begin{gathered}\begin{gathered}\\\\\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\red{ \bigstar} \: \underline{\bf{\orange{More \: Useful \: Formula}}}\\ {\boxed{\begin{array}{cc}\dashrightarrow \sf(a + b)^{2} = {a}^{2} + {b}^{2} + 2ab \\\\\dashrightarrow \sf(a - b)^{2} = {a}^{2} + {b}^{2} - 2ab \\\\\dashrightarrow \sf(a + b)(a - b) = {a}^{2} - {b}^{2} \\\\\dashrightarrow \sf(a + b) ^{3} = {a}^{3} + b^{3} + 3ab(a + b) \\\\ \dashrightarrow\sf(a - b) ^{3} = {a}^{3} - b^{3} - 3ab(a - b) \\ \\\dashrightarrow\sf a ^{3} + {b}^{3} = (a + b)(a ^{2} + {b}^{2} - ab) \\\\\dashrightarrow \sf a ^{3} - {b}^{3} = (a - b)(a ^{2} + {b}^{2} + ab \\\\\dashrightarrow \sf{a²+b²=(a+b)²-2ab}\\ \end{array}}}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}}}}}}}}}}}\end{gathered}\end{gathered}}\end{gathered}$