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Question

If $${x}^{4}+\cfrac{1}{{x}^{4}}=194$$, find $${x}^{3}+\cfrac{1}{{x}^{3}}$$, $${x}^{2}+\cfrac{1}{{x}^{2}}$$ and $$x+\cfrac{1}{x}$$


Solution

Given,

$$\Rightarrow x^4+\dfrac{1}{x^4}=194$$

$$\Rightarrow x^4+\dfrac{1}{x^4}+2=194+2$$

$$\Rightarrow \left ( x^2+\dfrac{1}{x^2} \right )^2=196$$

$$\therefore x^2+\dfrac{1}{x^2}=14$$.......$$(ii)$$

$$\Rightarrow x^2+\dfrac{1}{x^2}+2=14+2$$

$$\left ( x+\dfrac{1}{x} \right )^2=16$$

$$\therefore x+\dfrac{1}{x}=4$$.......$$(iii)$$

$$\Rightarrow \left ( x+\dfrac{1}{x} \right )^3=x^3+\dfrac{1}{x^3}+3\left ( x+\dfrac{1}{x} \right )$$

$$\Rightarrow 4^3=x^3+\dfrac{1}{x^3}+3(4)$$

$$\therefore x^3+\dfrac{1}{x^3} = 52$$........$$(i)$$

Maths

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