Solve the following

Solution:

Let us consider the following equation:

$x+\frac{1}{x}=9$

Squaring both sides, we get:

$\left(x+\frac{1}{x}\right)^{2}=(9)^{2}=81$

$\Rightarrow\left(x+\frac{1}{x}\right)^{2}=81$

$\Rightarrow x^{2}+2 \times x \times \frac{1}{x}+\left(\frac{1}{x}\right)^{2}=81$

$\Rightarrow x^{2}+2+\frac{1}{x^{2}}=81$

$\Rightarrow x^{2}+\frac{1}{x^{2}}=79$                (Subtracting 2 from both sides)

Now, squaring both sides again, we get:

$\left(x^{2}+\frac{1}{x^{2}}\right)^{2}=(79)^{2}=6241$

$\Rightarrow\left(x^{2}+\frac{1}{x^{2}}\right)^{2}=6241$

$\Rightarrow\left(x^{2}\right)^{2}+2\left(x^{2}\right)\left(\frac{1}{x^{2}}\right)+\left(\frac{1}{x^{2}}\right)^{2}=6241$

$\Rightarrow x^{4}+2+\frac{1}{x^{4}}=6241$

$\Rightarrow x^{4}+\frac{1}{x^{4}}=6239$