The image displays the expression $\sqrt[8]{16}$, which represents the eighth root of 16.

To solve this, we rewrite the expression using fractional exponents: $16^{\frac{1}{8}}$

Now, let's express 16 as a power of 2: $16 = 2^4$ Thus, we can rewrite $16^{\frac{1}{8}}$ as: $(2^4)^{\frac{1}{8}}$

Applying the rule $(a^m)^n = a^{m \cdot n}$: $2^{4 \cdot \frac{1}{8}} = 2^{\frac{4}{8}} = 2^{\frac{1}{2}}$

So, $2^{\frac{1}{2}} = \sqrt{2}$.

Therefore, the answer is: $\sqrt[8]{16} = \sqrt{2}$

Would you like further details or have any questions?

Here are 5 related questions to deepen understanding:

1. How do we simplify expressions with higher-order roots, like $\sqrt[16]{32}$?
2. What are fractional exponents, and how do they relate to roots?
3. Can you explain how to convert roots into fractional exponent form for different bases?
4. How would you solve $\sqrt[8]{256}$ using similar methods?
5. How does the power rule $(a^m)^n = a^{m \cdot n}$ apply in other situations?

Tip: Simplifying expressions often involves rewriting numbers as powers of smaller bases, which makes calculations more straightforward.