Question

If  \[x^4 + \frac{1}{x^4} = 194,\] then \[x^3 + \frac{1}{x^3} =\]

Options

• 76

• 52

• 64

• none of these

Answer 1

Given   `x^4 +1/x^4 = 194`

Using identity   `(a+b)^2 = a^2+2ab+b^2`

Here,  `a= x^2 , b = 1/x^2`

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

`(x^2 + 1/x^2 )^2 = x^4 +1/x^4 +2`

                           `(x^2+1/x^2)^2 = 194 +2`

                           `(x^2+1/x^2)^2 = 196`

       `(x^2+1/x^2)(x^2+1/x^2)^2 = 14 xx14`

                                  `x^2+1/x^2 = 14`

Again using identity  `(a+b)^2 = a^2 +2ab +b^2`

Here  `a=x,b=1/x`

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

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

Substituting  `x^2 +1/x^2 = 14`

`(x+1/x)^2 = 14 +2`

`(x+1/x)^2 = 16`

       `x+1/x = 4`

Using identity  `a^3 +b^3 = (a+b)(a^2 - ab +b^2)`

Here  `a= x^3, b= 1/x^3`

`x^3 +1/x^3 = (x+1/x)(x^2 - x xx 1/x+1/x^2)`

`x^3 +1/x^3 = (4)(-1 +14)`

`x^3 +1/x^3 = (4)(13)`

`x^3 +1/x^3 = 52`

Hence the value of   `x^3 +1/x^3`is  52.