Question

If x=9-4√5 then find x^6+1/x^6

Answer 1

Answer:

= > x = 9 - 4$\sqrt5$

= > 1/x = 1/9-4$\sqrt5$

By rationalization:

= > 1/x = (9+4$\sqrt5$)/(9-4$\sqrt5$)(9+4$\sqrt5$)

= > 1/x = (9+4$\sqrt5$)(81-80)

= > 1/x = 9 + 4$\sqrt5$

Therefore,

= > x + 1/x = 9-4$\sqrt5$ + 9 + 4$\sqrt5$

= > x + 1/x = 18

Square on both sides:

= > x^2+1/x^2+2=324

= > x^2+1/x^2=322

Cube on both sides:

= > x^6+1/x^6 +3(x^2+1/x^2) = 33386248

= > x^6 + 1/x^6 = 33385282

Answer 2

$\huge{\underline{\mathtt{\red{Solution:-}}}}$

$\star \: {\sf{\green{Given}}}$

$\star{\tt{If\: x \: = 9-4\sqrt{5}}}$

$\star \: {\sf{\blue{To \: Find:-}}}$

$\star{\tt{\frac{x^6+1}{x^6}}}$

${\bf{\underline{Solution:- }}}$

${\implies{\tt{x \: = 9- 4\sqrt{5}}}}$

${\implies{\tt{\frac{1}{x} =\frac{1} {9- 4\sqrt{5}}}}}$

By rationalizing,

${\implies{\tt{\frac{1}{x} = \frac{(9+4\sqrt{5})}{(9-4\sqrt{5})(9+4\sqrt{5})}}}}$

${\implies{\tt{\frac{1}{x} =\frac{1} {9+ 4\sqrt{5}}}}}$

NoW,

${\implies{\tt{\frac{x+1}{x} = \frac{9-4\sqrt{5}}{+9-4\sqrt{5}}}}}$

${\implies{\tt{\frac{x+1}{x} = 18}}}$

${\bf{\underline{By, \: Squaring \: both \: sides, \: We get ,}}}$

${\implies{\tt{\frac{x^2+1}{x^2+2} = 324}}}$

${\implies{\tt{\frac{x^2+1}{x^2} = 322}}}$

${\bf{\underline{By, \: Cubing \: both \: sides, }}}$

${\implies{\tt{\frac{x^6+1}{x^6+3}( \frac{x^2+1}{x^2}) = 33386248}}}$

${\implies{\tt{\frac{x^6+1}{x^6} = 33385282}}}$