Question

If x+1x=3 then, x7+1x7=?​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

GIVEN

$\displaystyle \sf{ x + \frac{1}{x} = 3\: }$

TO DETERMINE

$\displaystyle \sf{ {x}^{7} + \frac{1}{ {x}^{7} } \: }$

CALCULATION

$\displaystyle \sf{ x + \frac{1}{x} = 3\: }$

$\therefore \: \: \displaystyle \sf{ {x}^{2} + \frac{1}{ {x}^{2} } \: }$

$= \displaystyle \sf{ \bigg({ x + \frac{1}{x} \bigg)}^{2} - 2.x. \frac{1}{x} \: }$

$= \sf{ {3}^{2} - 2\: }$

$= \displaystyle \sf{ 9 - 2}$

$= \displaystyle \sf{ 7 }$

Again

$\: \: \displaystyle \sf{ {x}^{3} + \frac{1}{ {x}^{3} } \: }$

$= \displaystyle \sf{ \bigg({ x + \frac{1}{x} \bigg)}^{3} - 3.x. \frac{1}{x} .\bigg({ x + \frac{1}{x} \bigg)}\: }$

$= \displaystyle \sf{ {3}^{3} - 9\: }$

$= \displaystyle \sf{ 18\: }$

$\displaystyle \sf{ {x}^{4} + \frac{1}{ {x}^{4} } \: }$

$= \displaystyle \sf{ {\bigg( {x}^{2} + \frac{1}{ {x}^{2} } \bigg) }^{2} - 2. {x}^{2} . \frac{1}{ {x}^{2} } \: }$

$= \displaystyle \sf{ {7}^{2} - 2 \: }$

$= \displaystyle \sf{ 49 - 2 \: }$

$= \displaystyle \sf{ 47 \: }$

Now

$\: \: \displaystyle \sf{ \bigg( {x}^{3} + \frac{1}{ {x}^{3} } \bigg) \: \times }\displaystyle \sf{ \bigg( {x}^{4} + \frac{1}{ {x}^{4} } \bigg) = 47 \times 18 \: }$

$\implies \displaystyle \sf{ \bigg( {x}^{7} + \frac{1}{ {x}^{7} } \bigg) \: + }\displaystyle \sf{ \bigg( {x}^{} + \frac{1}{ {x}^{} } \bigg) = 846 \: }$

$\implies \displaystyle \sf{ \bigg( {x}^{7} + \frac{1}{ {x}^{7} } \bigg) \: + 3= 846 \: }$

$\implies \displaystyle \sf{ \bigg( {x}^{7} + \frac{1}{ {x}^{7} } \bigg) \: = 846 - 3 \: }$

$\implies \displaystyle \sf{ \bigg( {x}^{7} + \frac{1}{ {x}^{7} } \bigg) \: = 843\: }$

RESULT

$\boxed{ \displaystyle \sf{ \: \: \bigg( {x}^{7} + \frac{1}{ {x}^{7} } \bigg) \: = 843\: } \: \: \: }$

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If 2^(x+3) + 2^(x – 4) =129, then find the value of x.

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Answer 2

Answer

(x⁷+1/x⁷) = 843

• prove that (x⁷+1/x⁷) = 843