https://file.aiquickdraw.com/mathbot/file/1725891297336zi25thtp.jpg

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.

If a = 3 + 2√2, then find (a^6 + a^4 + a^2 + 1) / a^3?

This problem explores simplifying expressions involving radicals by using binomial expansion and algebraic identities. Learn step-by-step how to solve (a^6 + a^4 + a^2 + 1) / a^3 when a = 3 + 2√2. Suitable for advanced algebra students in Grades 11-12.

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