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

The given expression is:

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

Let's simplify this step by step.

### Step 1: Simplifying the numerator

The numerator \((2\sqrt{2} - 3a)(2\sqrt{2} + 3a)\) is a difference of squares, which follows the identity:

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

Here, \(a = 2\sqrt{2}\) and \(b = 3a\). Applying the identity:

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

Now calculate each square:

\[
(2\sqrt{2})^2 = 2^2 \times (\sqrt{2})^2 = 4 \times 2 = 8
\]
\[
(3a)^2 = 9a^2
\]

Thus, the numerator becomes:

\[
8 - 9a^2
\]

### Step 2: Writing the full expression

Now the expression becomes:

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

### Final result:

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

Would you like to explore further details or ask any questions about this expression?

Here are 5 related questions:
1. How can you simplify a binomial raised to a power, such as \((x + y)^3\)?
2. What is the difference of squares identity and how is it applied?
3. How do you simplify square roots in expressions like \(2\sqrt{2}\)?
4. What happens if you expand the denominator \(\left(\frac{x}{3} + 2y\right)^3\)?
5. How would this expression change if additional variables are introduced in the numerator?

**Tip**: When working with square roots and powers, always check if you can apply identities like difference of squares for faster simplification!

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

This guide provides a detailed step-by-step solution to simplify the expression (2√2 - 3a)(2√2 + 3a) ÷ (x/3 + 2y)^3 using algebraic identities like the difference of squares. Learn how to handle radicals and powers in algebra, suitable for students in Grades 9-12.

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