Sure, let's solve the problem step by step.
Given:
[tex]\[ a = 3 - 2\sqrt{2} \][/tex]
To find:
[tex]\[ a^4 - \frac{1}{a^4} \][/tex]
Step 1: Calculate [tex]\(a^4\)[/tex]
Starting with our expression for [tex]\(a\)[/tex],
[tex]\[ a = 3 - 2\sqrt{2} \][/tex]
Raise [tex]\(a\)[/tex] to the fourth power to find [tex]\(a^4\)[/tex]:
[tex]\[ a^4 = (3 - 2\sqrt{2})^4 \][/tex]
The calculated value for [tex]\(a^4\)[/tex] is:
[tex]\[ a^4 \approx 0.000866551777220085 \][/tex]
Step 2: Calculate [tex]\(\frac{1}{a^4}\)[/tex]
Next, we need to compute the reciprocal of [tex]\(a^4\)[/tex]:
[tex]\[ \frac{1}{a^4} \][/tex]
The calculated value for [tex]\(\frac{1}{a^4}\)[/tex] is:
[tex]\[ \frac{1}{a^4} \approx 1153.9991334482281 \][/tex]
Step 3: Find [tex]\(a^4 - \frac{1}{a^4}\)[/tex]
Finally, subtract [tex]\(\frac{1}{a^4}\)[/tex] from [tex]\(a^4\)[/tex]:
[tex]\[ a^4 - \frac{1}{a^4} \][/tex]
Using the values obtained:
[tex]\[ a^4 \approx 0.000866551777220085 \][/tex]
[tex]\[ \frac{1}{a^4} \approx 1153.9991334482281 \][/tex]
We get:
[tex]\[ a^4 - \frac{1}{a^4} \approx 0.000866551777220085 - 1153.9991334482281 \][/tex]
[tex]\[ a^4 - \frac{1}{a^4} \approx -1153.998266896451 \][/tex]
Therefore, the value of [tex]\(a^4 - \frac{1}{a^4}\)[/tex] is:
[tex]\[ \boxed{-1153.998266896451} \][/tex]
