Find the value

Here we have

$x = 3+2 \sqrt{2}$

$\implies \dfrac{1}{x} = 3 -2 \sqrt{2}$

Adding Both

$\implies x+ \dfrac{1}{x} = 6$

$\implies (x+\dfrac{1}{x})^2= 36$

$\implies (x^2 + \dfrac{1}{x^2} + 2) = 36$

$\implies (x^2 + \dfrac{1}{x^2})= 34$

$\implies (x^2 + \dfrac{1}{x^2} )^2 =34^2$

$\implies (x^4 + \dfrac{1}{x^4} +2) =34^2$

$\implies (x^4 + \dfrac{1}{x^4}) =1156-2 = \boxed{1154}$