If x^4 + 1/x^4 = 727, then find the value of x^3 - 1/x^3

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`x^4+1/x^4=727`

`x^4+1/x^4+2=727+2`

`(x^2+1/x^2)^2=(27)^2`

` ` Taking square roots both side

`x^2+1/x^2=27`

`x^2+1/x^2-2=27-2`

`(x-1/x)^2=(5)^2`

Taking square root again

`x-1/x=5`

` ` raise power 3 both side

`(x-1/x)^3=(5)^3`

` ` Expand left side ,we have

`x^3-1/x^3-3(x)(1/x)(x-1/x)=125`

` ` `x^3-1/x^3-3xx5=125`

` ` `x^3-1/x^3=125+15`

`x^3-1/x^3=140`

If `x-1/x=-5`

`x^3-1/x^3-3(x)(1/x)(x-1/x)=-125`

`x^3-1/x^3=-125-15`

`x^3-1/x^3=-140`

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