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.