Question

if x+1/x=6,find x^5+1/x^5

Answer 1

x+1/x=6 ……………..(1)

Squaring both sides

x^2+1/x^2+2=36

x^2+1/x^2=34 ………..(2)

On cubing eq.(1) both sides

x^3+1/x^3+3.x1/x.(x+1/x)=216

x^3+1/x^3+3×6=216

x^3+1/x^3=216–18

x^3+1/x^3=198………………(3)

Multiply eq.(2) & (3)

(x^2+1/x^2)×(x^3+1/x^3)=34×198

x^5+x+1/x+1/x^5=6732

x^5+6+1/x^5=6732

x^5+1/x^5=6732–6=6726 , Answer

Answer 2

Hope u like my process
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$= > x + \frac{1}{x} = 6 \\ \\ \underline{ \bf \: \: squaring \: \: both \: \: sides \: \: } \\ \\ = > {x}^{2} + \frac{1}{ {x}^{2} } + 2x \frac{1}{x} = {6}^{2} \\ \\ = > {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 36 \\ \\ = > \boxed{ \bf {x}^{2} + \frac{1}{ {x}^{2} } = 36 - 2 = \underline{ \green{34} }} \\ \\ now \\ \\ again \: \: x + \frac{1}{x} = 6 \\ \\ \underline{ \bf \: \: cubing \: \: both \: \: sides \: \: } \\ \\ = > {x}^{3} + \frac{1}{ {x}^{3} } + 3 {x}\frac{1}{x} (x + \frac{1}{ {x} }) = {6}^{3} \\ \\ = > {x}^{3} + \frac{1}{ {x}^{3} } + 3 \times 1 \times 6 = 216 \\ \\ = > \boxed{ \bf \: {x}^{3} + \frac{1}{ {x}^{3} } = 216 - 18 = \underline{ \green{198}} }$
Now..

Multiplying both equation we get..

$= > ( {x}^{2} + \frac{1}{ {x}^{2} } )( {x}^{3} + \frac{1}{ {x}^{3} } ) = 34 \times 198 \\ \\ = > {x}^{2 + 3} + \frac{1}{ {x}^{2 + 3} } + {x}^{3 - 2} + \frac{1}{ {x}^{3 - 2} } = 6732 \\ \\ = > {x}^{5} + \frac{1}{ {x}^{5} } + (x + \frac{1}{x} )= 6732 \\ \\ = > \boxed{ \bf \: {x}^{5} + \frac{1}{ {x}^{5} } = 6732 - 6 = \underline{ \orange{6726}} }$
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