Let's solve the expression given in the image:

[ 2^3 \sqrt[4]{ \sqrt[3]{3} + \sqrt[16]{16} } ]

We'll start by simplifying the components inside the expression:

1. Calculating the cube root and the fourth root:

   • $\sqrt[3]{3}$ remains as is, since it's already in its simplest form.
   • $\sqrt[16]{16} = 16^{1/16} = 2$ because $16 = 2^4$ and $(2^4)^{1/16} = 2^{4/16} = 2^{1/4}$.

2. Adding the results:

   • $\sqrt[3]{3} + \sqrt[16]{16} = \sqrt[3]{3} + 2$

3. Calculating the fourth root:

   • We need to compute $\sqrt[4]{\sqrt[3]{3} + 2}$. This value cannot be simplified further without numerical approximation.

4. Multiplying by $2^3$:

   • $2^3 = 8$, so the expression becomes $8 \sqrt[4]{\sqrt[3]{3} + 2}$.

Thus, the simplified version of the expression is: [ 8 \sqrt[4]{\sqrt[3]{3} + 2} ]

This expression cannot be simplified further analytically without using numerical methods to approximate $\sqrt[3]{3}$.

Would you like details on how any part of this calculation was done or have any other questions?

1. How can we approximate $\sqrt[3]{3}$ using a calculator?
2. What are the properties of roots that were used in this calculation?
3. How could this expression be used in a real-world application?
4. Could we graph the function $\sqrt[4]{x + 2}$ to better understand its behavior?
5. Are there other similar mathematical expressions you might need help with?

Tip: Always double-check that the operations inside and outside the roots and powers are done in the correct sequence to avoid errors in calculation.