The mathematical problem in the image appears to involve nested radicals and a product of square roots. I will transcribe the expression and solve it step by step.

The expression appears to be:

[ \sqrt{\frac{\sqrt{3} - 3}{\sqrt[3]{8}} + 4.5\sqrt[3]{5}} \cdot \sqrt[3]{2} ]

Solution:

Step 1: Simplify each component.

1. $\sqrt[3]{8} = 2$, since $2^3 = 8$.
2. Rewrite the fraction: $\frac{\sqrt{3} - 3}{\sqrt[3]{8}} = \frac{\sqrt{3} - 3}{2}.$
3. Rewrite $4.5 \sqrt[3]{5}$ as-is since it doesn't simplify further: $4.5 \sqrt[3]{5}.$

Step 2: Combine terms inside the square root.

The expression under the square root becomes: $\sqrt{\frac{\sqrt{3} - 3}{2} + 4.5\sqrt[3]{5}}.$

Step 3: Multiply by $\sqrt[3]{2}$.

The final expression after including the product becomes: [ \sqrt{\frac{\sqrt{3} - 3}{2} + 4.5\sqrt[3]{5}} \cdot \sqrt[3]{2}. ]

Step 4: Evaluate numerically (optional).

If you want, I can evaluate this numerically, but the expression above is its simplified form.

Would you like to proceed with a numerical evaluation, or do you want further clarification on any part of this process?

Related Questions:

1. What are the properties of cube roots that differ from square roots?
2. How do you simplify nested radicals effectively?
3. How do rational and irrational numbers behave under addition and multiplication?
4. Can this expression be simplified further without numerical computation?
5. What is the significance of radicals in solving equations?

Tip:

Always simplify individual components of an expression before combining them to make calculations more manageable!