Question

if m-2/m=5 show m^4+1/m^4=727​

Answer 1

➲CORRECT QUESTION :

If $\bf{m - \dfrac{1}{m} = 5 }$

Show that $\bf{ m^4 + \dfrac{1}{ m^4} = 727}$.

☯GIVEN :

• $\bf{m - \dfrac{1}{m} = 5}$.

☯TO FIND :

• Show that $\bf{ m^4 + \dfrac{1}{ m^4} = 727}$.

☯FORMULA REQUIRED :

• $\bigstar \underline{ \boxed{ \red{\bf{{(a -b)}^{2} = a^2 - 2ab + b^2 }}}}$

• $\bigstar \underline{ \boxed{ \red{\bf{{(a +b)}^{2} = a^2 +2ab + b^2 }}}}$

➲SOLUTION :

• Squaring on both sides :

$\implies { \sf{{ \bigg(m - \dfrac{1}{m} \bigg)}^{2} = (5) ^2 }}$

$\implies { \sf{ {(m)}^{2} - 2 \times \cancel{m} \times \dfrac{1}{ \cancel{m}} + { \bigg( \dfrac{1}{m} \bigg )}^{2} = 25 }}$

$\implies { \sf{ {m}^{2} - 2 + \dfrac{1}{ m^2 } = 25}}$

$\implies { \sf{ {m}^{2} + \dfrac{1}{ m^2 } = 25 + 2 }}$

$\implies \underline{\boxed{ \bf{ {m}^{2} + \dfrac{1}{ {m}^{2} } = 27}}}$

• Squaring on both sides (Again) :

$\implies { \sf{{ \bigg(m^2 + \dfrac{1}{m^2} \bigg)}^{2} = (27)^2 }}$

$\implies { \sf{ {(m^2)}^{2} + 2 \times \cancel{m^2} \times \dfrac{1}{ \cancel{m^2}} + { \bigg( \dfrac{1}{m^2} \bigg )}^{2} =729 }}$

$\implies { \sf{ {m}^{4} + 2 + \dfrac{1}{ m^4 } = 729 }}$

$\implies { \sf{ {m}^{4} + \dfrac{1}{ m^4 } = 729 - 2 }}$

$\implies \underline{\boxed{ \pink {\bf{ {m}^{4} + \dfrac{1}{ m^4} = 727}}}}$

$\huge{\green{\therefore}}$$\bf{ m^4 + \dfrac{1}{ m^4} =727 }$.