Question

If 3x - 1/3x=5 find 81x^4 + 1/81x^4​

Answer 1

Step-by-step explanation:

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Answer 2

Step-by-step explanation:

$\bf \underline{★Given-}$

$\rm{3x - \frac{1}{3x} = 5}$

$\bf \underline{★To \: find-}$

$\rm{the \: value \: of \: :81 {x}^{4} + \frac{1}{81 {x}^{4} } = \: ? }$

$\bf \underline{★Solution-}$

$\textsf{We have,}$

$\rm{3x - \frac{1}{3x} = 5}$

$\textsf{Squaring on both sides, we get}$

$\rm{ \bigg(3x - \frac{1}{3x} \bigg) ^{2} = (5 {)}^{2} }$

$\textsf{Now, comparing the given expression with (a-b)², we get}$

$\: \: \: \: \: \rm{a = 3x \: and \: b = \frac{1}{3x} }$

$\textsf{Using bionomial identity (a-b)² = a²-2ab+b², we have}$

$\rm{ \bigg(3x - \frac{1}{3x} \bigg) ^{2} = (5 {)}^{2} }$

$\rm{\implies \: (3x {)}^{2} - 2(3x) \bigg( \frac{1}{3x} \bigg) + \bigg(\frac{1}{3x} \bigg)^{2} = 25 }$

$\rm{\implies \:9x^{2} - 2( \cancel{3x}) \bigg( \frac{1}{ \cancel{3x}} \bigg) + \frac{1}{9 {x}^{2} } = 25}$

$\rm{\implies \:9 {x}^{2} - 2 + \frac{1}{9x^{2} } = 25}$

$\rm{\implies \:9 {x}^{2} + \frac{1}{9 {x}^{2} } = 25 + 2 }$

$\rm{\implies \:9 {x}^{2} + \frac{1}{9 {x}^{2} } = 27 }$

$\textsf{★Again, squaring on both sides, we get}$

.

$\rm{ \bigg(9 {x}^{2} + \frac{1}{9 {x}^{2} } \bigg)^{2} = (27 {)}^{2} }$

$\textsf{Now, comparing the given expression with (a+b)², we get}$

$\: \: \: \: \: \: \rm{a = 9 {x}^{2} \: and \: b = \frac{1}{9 {x}^{2} } }$

$\textsf{Using bionomial identity (a+b)² = a²+2ab+b², we have}$

$\rm{ \bigg(9 {x}^{2} + \frac{1}{9 {x}^{2} } \bigg)^{2} = (27 {)}^{2} }$

$\rm{ \implies(9 {x}^{2} {)}^{2} + 2(9 {x}^{2}) \bigg( \frac{1}{9 {x}^{2} } \bigg) + \bigg( \frac{1}{9 {x}^{2} } \bigg )^{2} } = (27 {)}^{2}$

$\rm{ \implies81 {x}^{4} + 2( \cancel{9 {x}^{2}} ) \bigg( \frac{1}{ \cancel{9 {x}^{2}} } \bigg) + \frac{1}{81 {x}^{4} } = 729}$

$\rm{ \implies81 {x}^{4} + 2 + \frac{1}{81 {x}^{4} } = 729}$

$\rm{ \implies81 {x}^{4} + \frac{1}{81 {x}^{4} } = 729 - 2}$

$\rm{ \implies81 {x}^{4} + \frac{1}{81 {x}^{4} } = 727 }$

$\bf{Hence, the \: value \: of \: : 81 {x}^{4} + \frac{1}{81 {x}^{4} } \: is \: 727.}$