Question

plz ans the question is attached​

Answer 1

☞︎︎︎ Solution :

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★ Given :

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$\large \: \: \: \: \rm \rightarrow \: {x}^{2} + \dfrac{1}{ {x}^{2} } = 51$

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☆ On Solving -

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• Identity

➪ (a - b)² = a² + b² - 2ab

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$\large \rm \: : ⇒ {\left( {x - \dfrac{1}{ {x}^{} } }^{} \right) }^{2} = {x}^{2} + \dfrac{1}{ {x}^{2} } - 2 \times x \times \dfrac{1}{x}$

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$\large \rm \: : ⇒ {\left( {x - \dfrac{1}{ {x}^{} } }^{} \right) }^{2} = {x}^{2} + \dfrac{1}{ {x}^{2} } - 2 \times \cancel{x} \times \dfrac{1}{ \cancel{x}}$

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$\large \rm \: : ⇒ {\left( {x - \dfrac{1}{ {x}^{2} } }^{} \right)}^{2} = {x}^{2} + \dfrac{1}{ {x}^{2} } - 2$

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$\large \: \: \: \: \rm \rightarrow \: {x}^{2} + \dfrac{1}{ {x}^{2} } = 51$

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$\large \rm \: : ⇒ {\left( {x - \dfrac{1}{ {x}^{} } }^{} \right) }^{2} = 51 - 2$

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$\large \rm \: : ⇒{ \left( {x - \dfrac{1}{ {x}^{} } }^{} \right) }^{2} = 49$

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$\large \rm \: : ⇒{ \left( {x - \dfrac{1}{ {x}^{} } }^{} \right) }^{} = \sqrt{ 49}$

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$\large \rm \: : ⇒{ \left( {x - \dfrac{1}{ {x}^{} } }^{} \right) }^{} = 7$

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• On cubicing both sides -

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$\: \: \: \: \: \: \: \large \rm \: : ⇒{ \left( {x - \dfrac{1}{ {x}^{} } }^{} \right) }^{3} = {(7) }^{3}$

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☆ Using Identity :

→ a³ - b³ = a³ - b³ - 3(a + b)

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$\large \rm : ⇒ \: {{{x }^{3} - \dfrac{1}{ {x}^{3} } }^{} } - 3 \left(x - \dfrac{1}{x} \right) = 343$

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$\large \rm : ⇒ \: {{{x }^{3} - \dfrac{1}{ {x}^{3} } }^{} } - 3(7) = 343$

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$\large \rm : ⇒ \: {{{x }^{3} - \dfrac{1}{ {x}^{3} } }^{} } - 21= 343$

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• On Transposing The Terms :

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$\large \rm : ⇒ \: {{{x }^{3} - \dfrac{1}{ {x}^{3} } }^{} } = 343 + 21$

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$\: \: \: \boxed{ \large \rm \therefore \: \: {{{x }^{3} - \dfrac{1}{ {x}^{3} } }^{} } = 364}$