Question

x+1/x=5, then find the value of x^16+1/x^16 ?

Answer 1

Given

$x+1/x=5$

Tahing square on both sides we have

$(x+1/x)^2=5^2$

solving the bracket by using formula $(a+b)^2=a^2+2ab+b^2$

we get

$x^2+2(x)(1/x)+1/x^2=25,$

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

$=x^2+1/x^2=23$

again taking aquare on both sides we have ,

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

$=x^4+(2)(x^2)(1/x^2)+1/x^4=529,$

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

$=x^4+1/x^4=527$

taking aquare on both sides we have

$=(x^4+1/x^4)^2=(527)^2$

$=x^8+(2)(x^4)(1/x^4)+1/x^8=277729,$

$x^8+(2)+1/x^8=277729-2$

$=x^8+1/x^8=277727$

taking aquare on both sides we have

$=(x^8+1/x^8)^2=(277727)^2$

$=x^{16}+(2)(x^8)(1/x^8)+1/x^{16}=(277727)^2,$

$x^{16}+1/x^{16}=(277727)^2-2$

Answer:

$x^{16}+1/x^{16}=(277727)^2-2$