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The inequality given is:

\[
2 < \sqrt[3]{x} < \sqrt{5}
\]

To solve for \( x \), follow these steps:

### Step 1: Cube each part of the inequality
Cubing the entire inequality will remove the cube root on \( \sqrt[3]{x} \).

\[
2^3 < x < (\sqrt{5})^3
\]

Simplifying each term:

\[
8 < x < \sqrt{5}^3
\]

We need to calculate \( \sqrt{5}^3 \), which is:

\[
\sqrt{5}^3 = 5^{3/2} = 5 \times \sqrt{5} \approx 11.18
\]

Thus, the inequality becomes:

\[
8 < x < 11.18
\]

### Final Answer:
The solution for \( x \) is:

\[
8 < x < 11.18
\]

Would you like further details or have any questions?

### Follow-up Questions:
1. How do you simplify cube roots in inequalities?
2. What happens when you square or cube both sides of an inequality?
3. How does \( \sqrt[3]{x} \) behave differently from \( \sqrt{x} \)?
4. How do you find approximate values for cube roots?
5. Can you explain the properties of fractional exponents?

### Tip:
Always check if the operations you perform on inequalities are valid across the entire number range, especially when dealing with square and cube roots.

Solve the inequality: 2 < cube root of x < square root of 5. Find x.

Learn how to solve the inequality 2 < cube root of x < square root of 5. This problem covers key mathematical concepts like cube roots, square roots, exponents, and solving inequalities. Detailed step-by-step solution provided for Grades 9-10.

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