To find the value of [tex]\( x^4 + \frac{1}{x^4} \)[/tex] for [tex]\( x = \sqrt{5} + 2 \)[/tex], we need to follow a series of steps to evaluate each component and then combine them.
1. Determine [tex]\( x \)[/tex]:
[tex]\[
x = \sqrt{5} + 2
\][/tex]
2. Calculate [tex]\( x^4 \)[/tex]:
Elevate [tex]\( x \)[/tex] to the power of 4.
[tex]\[
x^4 = (\sqrt{5} + 2)^4
\][/tex]
When evaluated, [tex]\( x^4 \approx 321.9968943799849 \)[/tex].
3. Calculate [tex]\( \frac{1}{x^4} \)[/tex]:
Since [tex]\( x = \sqrt{5} + 2 \)[/tex], evaluate [tex]\(\frac{1}{x^4}\)[/tex].
[tex]\[
\frac{1}{x^4} \approx 0.003105620015141858
\][/tex]
4. Add [tex]\( x^4 \)[/tex] and [tex]\( \frac{1}{x^4} \)[/tex]:
Combine the results obtained in steps 2 and 3.
[tex]\[
x^4 + \frac{1}{x^4} \approx 321.9968943799849 + 0.003105620015141858
\][/tex]
[tex]\[
x^4 + \frac{1}{x^4} \approx 322.00000000000006
\][/tex]
Thus, the value of [tex]\( x^4 + \frac{1}{x^4} \)[/tex] for [tex]\( x = \sqrt{5} + 2 \)[/tex] is approximately [tex]\( 322.00000000000006 \)[/tex].
