The given expression in the image is:

$\frac{a^6 + a^4 + a^2 + 1}{a^3}$

with $a = 3 + 2\sqrt{2}$.

Step-by-Step Solution:

1. Let $a = 3 + 2\sqrt{2}$. We will find powers of $a$ like $a^2$, $a^3$, etc.

   First, calculate $a^2$: $a^2 = (3 + 2\sqrt{2})^2 = 9 + 12\sqrt{2} + 8 = 17 + 12\sqrt{2}$

2. Now calculate $a^3$: $a^3 = a \times a^2 = (3 + 2\sqrt{2}) \times (17 + 12\sqrt{2})$ Expanding this: $a^3 = 3 \times 17 + 3 \times 12\sqrt{2} + 2\sqrt{2} \times 17 + 2\sqrt{2} \times 12\sqrt{2}$ $a^3 = 51 + 36\sqrt{2} + 34\sqrt{2} + 48 = 99 + 70\sqrt{2}$

3. Next, we need to compute $a^6$, $a^4$, and $a^2$. Notice the pattern from powers of binomials, so it is more practical to look for a possible simplification of the original expression: $\frac{a^6 + a^4 + a^2 + 1}{a^3}$

I will simplify this expression using symbolic algebra to look for common factors and evaluate it for the given $a$.The value of the given expression $\frac{a^6 + a^4 + a^2 + 1}{a^3}$, when $a = 3 + 2\sqrt{2}$, is 204.

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. What are other methods to simplify expressions involving radicals?
2. How can binomial expansion help in finding higher powers of binomial terms?
3. How would the value change if a different radical form was used for $a$?
4. What techniques can be used to compute powers of terms like $a = 3 + 2\sqrt{2}$ efficiently?
5. How can symbolic algebra systems help in solving complex algebraic expressions?

Tip: When dealing with expressions that involve powers of binomials, look for patterns or shortcuts like using binomial expansion or factoring techniques to simplify the work.