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Instead of trying to figure out the value of x, we can use clever squaring.

Square the first equation.

\((x+\frac{1}{x})^2= 3^2\)

\( x^2+1+1+\frac{1}{x^2}=9\)

\(x^2+\frac{1}{x^2}=7\)

Square this equation again.

\((x^2+\frac{1}{x^2})^2=7^2\)

\(x^4+1+1+\frac{1}{x^4}=49\)

\(x^4+\frac{1}{x^4}=47\)