Question

x^4 +1/x^4 = 119 then the value of x^3 -1/x^3

Answer 1

x⁴ + 1/x⁴ = 119
=> x⁴ + 1/x⁴ + 2 = 121
=> (x² + 1/x²) ² = 11²
=> x² + 1/x² = 11    ,  ignore the negative value as LHS is +ve.

=> x² + 1/x² - 2 = 9
=> (x - 1/x)² = 3²
=> x - 1/x = +3 or -3

=> (x - 1/x)³ = x³ - 1/x³ - 3 x * 1/x * (x - 1/x)
      (+ 3) ³ = x³ - 1/x³ - 3 ( + 3)

Answer:      x³ - 1/x³ = 3³ + 9 or -30 - 9 = + 36 

Answer 2

${x}^{4} + \frac{1}{ {x}^{4} } = 119$

Adding 2 on both sides;

x^4 + 1/x^4 +2 = 119 +2

(x²+1/x²)²=121

x²+1/x²=11

Adding -2 on both sides

x²-2+1/x²=9

(x-1/x)²= 9

x-1/x= ±3

Now, (x-1/x)³= x³-1/x³-3(x-1/x)


Consider x-1/x= 3 .
3³ = x³-1/x³-3(3)

27= x³-1/x³ - 9

27+9=x³-1/x³

36 = x³-1/x³

consider x-1/x = -3

-27 = x³-1/x³-3(-3)

-27 = x³-1/x³+9


-27-9 = x³-1/x³

x³-1/x³ = ±36