-step explanation:Given :- x+(1/x) = 5To find :-Find the value of x⁹+(1/x⁹) ?Solution :-Given that :x+(1/x) = 5 ------------(1)On cubing both sides[x+(1/x)]³ = 5³=> [x+(1/x)]³ = 125We KNOW that(a+b)³ = a³+b³+3ab(a+b)Where a = x and b = 1/xNow,=>x³+(1/x)³+3(x)(1/x)[x+(1/x)] = 125=> x³+(1/x³) +3(x/x)[x+(1/x)] = 125=> x³+(1/x³) +3(1)[x+(1/x)] = 125=>x³+(1/x³) +3[x+(1/x)] = 125=> x³+(1/x³)+3(5) = 125 (from (1))=> x³+(1/x³)+ 15= 125=> x³+(1/x³) = 125-15=> x³+(1/x³) = 110 ----------(2)On cubing both sides then=> [x³+(1/x³)]³ = (110)³We know that(a+b)³ = a³+b³+3ab(a+b)Where a = x³ and b = 1/x³Now,=>(x³)³+(1/x³)³+3(x³)(1/x³)[x³+(1/x³)] = 1331000=> x⁹+(1/x⁹)+3(x³/x³)[x³+(1/x³)] = 1331000=> x⁹+(1/x⁹)+3(1)[x³+(1/x³)] = 1331000=> x⁹+(1/x⁹)+3[x³+(1/x³)] = 1331000=> x⁹+(1/x⁹)+3(110) = 1331000=> x⁹+(1/x⁹) +330 = 1331000=> x⁹+(1/x⁹) = 1331000-330=> x⁹+(1/x⁹) = 1330670Answer :-The value of x⁹+(1/x⁹) for the given problem is 1330670Used FORMULAE:-(a+b)³ = a³+b³+3ab(a+b)(a^m)^N = a^(mn)