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The given expression is:

\[
E = \sqrt[3]{3} + \sqrt{8} \cdot \left( \sqrt[3]{3} - \sqrt{8} \right)
\]

Let's break this down step by step.

### Step 1: Simplifying each component

- \(\sqrt[3]{3}\) is the cube root of 3. This is an irrational number, but we can keep it as \(\sqrt[3]{3}\) for now.
  
- \(\sqrt{8} = 2\sqrt{2}\), since \(\sqrt{8}\) simplifies to \(2\sqrt{2}\).

So the expression becomes:

\[
E = \sqrt[3]{3} + 2\sqrt{2} \cdot \left( \sqrt[3]{3} - 2\sqrt{2} \right)
\]

### Step 2: Applying distribution

Distribute \(2\sqrt{2}\) across the terms in the parentheses:

\[
E = \sqrt[3]{3} + 2\sqrt{2} \cdot \sqrt[3]{3} - 2\sqrt{2} \cdot 2\sqrt{2}
\]

This becomes:

\[
E = \sqrt[3]{3} + 2\sqrt{2} \cdot \sqrt[3]{3} - 4 \cdot 2
\]

Since \(2\sqrt{2} \cdot \sqrt[3]{3}\) is not easily simplified, we'll leave it as is for now. The last term simplifies to \(-8\), so we have:

\[
E = \sqrt[3]{3} + 2\sqrt{2} \cdot \sqrt[3]{3} - 8
\]

### Final Expression

So the simplified expression for \(E\) is:

\[
E = \sqrt[3]{3} + 2\sqrt{2} \cdot \sqrt[3]{3} - 8
\]

This expression contains both irrational numbers, so further simplification would require numerical approximations.

Would you like to compute this numerically or discuss any step in more detail? 

Here are some related questions to consider:
1. What are the steps for simplifying cube roots with coefficients?
2. How can we approximate irrational values like \(\sqrt[3]{3}\) and \(\sqrt{2}\)?
3. How does distributing radicals work in expressions like this?
4. What properties of radicals make terms like \(2\sqrt{2}\) and \(\sqrt{8}\) equivalent?
5. Can this expression be simplified further using approximate values?

**Tip:** For expressions with cube roots and square roots, it's helpful to keep terms symbolic until you're ready to compute approximations.

Learn how to simplify the expression E = ³√3 + √8 · (³√3 − √8). This problem covers key algebraic concepts including cube roots, square root simplification, and the distributive property. Suitable for students in Grades 9-12.

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