find the values of

Question:

If $\left(x-\frac{1}{x}\right)=5$, find the values of

(i) $\left(x^{2}+\frac{1}{x^{2}}\right)$

(ii) $\left(x^{4}+\frac{1}{x^{4}}\right)$.

Solution:

$(i)\left(x-\frac{1}{x}\right)=5$

$\Rightarrow$ Squaring both the sides:

$\Rightarrow\left(x-\frac{1}{x}\right)^{2}=(5)^{2}$

$\Rightarrow\left(x^{2}+\frac{1}{x^{2}}-2(x)\left(\frac{1}{x}\right)\right)=25$

$\Rightarrow\left(x^{2}+\frac{1}{x^{2}}\right)-2=25$

$\Rightarrow\left(x^{2}+\frac{1}{x^{2}}\right)=25+2$

$\Rightarrow\left(x^{2}+\frac{1}{x^{2}}\right)=27$

Therefore, the value of $\left(x^{2}+\frac{1}{x^{2}}\right)$ is 27 .

$\left(x^{2}+\frac{1}{x^{2}}\right)=27$

$\Rightarrow$ Squaring both the sides:

$\Rightarrow\left(x^{4}+\frac{1}{x^{4}}-2\left(x^{2}\right)\left(\frac{1}{x^{2}}\right)\right)=(27)^{2}$

$\Rightarrow\left(x^{4}+\frac{1}{x^{4}}\right)-2=729$

$\Rightarrow\left(x^{4}+\frac{1}{x^{4}}\right)=729+2$

$\Rightarrow\left(x^{4}+\frac{1}{x^{4}}\right)=731$

Therefore, the value of $\left(x^{4}+\frac{1}{x^{4}}\right)$ is 731 .

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