Question

if x-1/x=$\sqrt{5}$, find the value of x^4+1/x^4

Answer 1

Answer:

527

details

( x + 1/x) = 5

( x + 1/x)^2 =25

x^2 + 2 +1/x^2 =25

x^2 + 1/x^2 = 23

( x^2 + 1/x^2)^2 = 23^2 = 529

x^4 +2 + 1/x^4 = 529

x^4 +1/x^4 = 52

Answer 2

$x - \frac{1}{x} = \sqrt{5}$

${x}^{2} - \sqrt{5} x - 1 = 0$

$\implies x = { \phi}^{2} , - \frac{1}{ \phi ^{2} }$

$\phi = \frac{ \sqrt{5} + 1 }{4}$

Thus,

$\implies {x}^{4} + \frac{1}{ {x}^{4} } = { \phi}^{4} + { \phi}^{ - 4}$

$= 3\phi + 2+ (1 - \phi) {}^{4}$

$= (3 \phi + 2 )2+ 7 - 6 \phi - 4$

$= \cancel{6 \phi} +\cancel{ 4} + 7 - \cancel{6 \phi }- \cancel4$

$= 7$