Question

If x^2+1/x^2 =11,find the value of x-1/x​ and x^4+1/x^4​

Answer 1

Answer :-

➩ $\sf x^2 + \frac{1}{x^2} = 11$

Subtracting 2 from both side -

➩ $\sf x^2 + \frac{1}{x^2} - 2 = 11 - 2$

➩ $\sf (x)^2 + (\frac{1}{x^2}) - 2 ( x ) \times \frac{1}{x} = 9$

By using the identity -

• $\sf (a - b)^2 = a^2 + b^2 - 2ab$

➩ $\sf (x - \frac{1}{x})^2 = 9$

➩ $\boxed{\sf x - \frac{1}{x} = 3 }$

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⠀

➩ $\sf x^2 + \frac{1}{x^2} = 11$

Squaring on both side -

➩ $\sf (x^2 + \frac{1}{x^2})^2 = 11^2$

By using the identity -

• $\sf (a + b)^2 = a^2 + b^2 + 2ab$

➩ $\sf (x^2)^2 + (\frac{1}{x^2})^2 + 2 \times \cancel{x^2} \times \frac{1}{\cancel{x^2}} = 121$

➩ $\sf x^4 + \frac{1}{x^4} + 2 = 121$

➩ $\sf x^4 + \frac{1}{x^4} = 121 - 2$

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