Sure, let's solve this step-by-step.
Given [tex]\( x = \sqrt{5} + 2 \)[/tex], we need to find the value of [tex]\( x^4 + \frac{1}{x^4} \)[/tex].
Firstly, we calculate [tex]\( x^4 \)[/tex]. Given that [tex]\( x = \sqrt{5} + 2 \)[/tex]:
[tex]\[ x = \sqrt{5} + 2 \][/tex]
The next step is to raise this value to the power of 4:
[tex]\[ x^4 \approx 321.9968943799849 \][/tex]
Now, we need to find [tex]\( \frac{1}{x^4} \)[/tex]:
[tex]\[ \frac{1}{x^4} \approx 0.003105620015141858 \][/tex]
Finally, we add these two results together:
[tex]\[ x^4 + \frac{1}{x^4} \approx 321.9968943799849 + 0.003105620015141858 \][/tex]
Adding these values, we get:
[tex]\[ x^4 + \frac{1}{x^4} \approx 322.00000000000006 \][/tex]
So the value of [tex]\( x^4 + \frac{1}{x^4} \)[/tex] is approximately [tex]\( 322.00000000000006 \)[/tex].
