(2 radical din 2 -3a) (2 radical din 2 +3a)÷(x/3+2y)³

You provided the following expression:

\[
\frac{(2\sqrt{2} - 3a)(2\sqrt{2} + 3a)}{\left(\frac{x}{3} + 2y\right)^3}
\]

Let's break this down step-by-step.

### Step 1: Simplify the numerator
The numerator is a difference of squares, which follows the formula:

\[
(a - b)(a + b) = a^2 - b^2
\]

In your case:
- \( a = 2\sqrt{2} \)
- \( b = 3a \)

Applying the formula:

\[
(2\sqrt{2})^2 - (3a)^2
\]

Simplify each term:
- \( (2\sqrt{2})^2 = 4 \times 2 = 8 \)
- \( (3a)^2 = 9a^2 \)

So, the simplified numerator is:

\[
8 - 9a^2
\]

### Step 2: Denominator stays the same
The denominator is:

\[
\left(\frac{x}{3} + 2y\right)^3
\]

### Step 3: Final expression
After simplifying, the expression becomes:

\[
\frac{8 - 9a^2}{\left(\frac{x}{3} + 2y\right)^3}
\]

Would you like further details on any part of this solution?

### 5 Relative Questions:
1. Can you verify what range of values for \( a \) or \( x, y \) will simplify the expression?
2. How does this expression behave when \( x = 0 \) or \( y = 0 \)?
3. Would this be easier to work with if the denominator was expanded or simplified further?
4. What are the domain restrictions for \( x, y, a \) to keep this expression defined?
5. How would this expression change if powers or roots were involved in the denominator?

### Tip:
Always remember to check for domain restrictions when variables appear in the denominator!

(2 radical din 2 -3a) (2 radical din 2 +3a) ÷ (x/3 + 2y)^3

Explore the process of simplifying the expression (2√2 - 3a)(2√2 + 3a) ÷ (x/3 + 2y)^3 using the difference of squares theorem and exponentiation rules. This detailed solution is ideal for high school students in Grades 9-12 studying algebra.

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