Solve Nested Radicals: Fourth Root of Cube Root of 16

Solve the following exercise:

$\sqrt[4]{\sqrt[3]{16}}=$

To solve this problem, we'll convert the roots into exponents and simplify:

• Step 1: Express the nested radicals in terms of exponents.
• Step 2: Simplify by multiplying the exponents.
• Step 3: Compare the simplified result to the given choices.

Now, let's work through each step:
Step 1: The cube root of 16 can be written as $16^{\frac{1}{3}}$. Thus, our expression becomes $\sqrt[4]{16^{\frac{1}{3}}}$.
Step 2: Apply the fourth root, which is an exponent of $\frac{1}{4}$. This gives us $(16^{\frac{1}{3}})^{\frac{1}{4}} = 16^{\frac{1}{3} \cdot \frac{1}{4}} = 16^{\frac{1}{12}}$.
Step 3: From the original question, the expression simplifies to $16^{\frac{1}{12}}$, which is equivalent to $\sqrt[12]{16}$. Therefore, the choices that are correct are the ones that reflect this equivalence.

Therefore, the solution to the problem involves recognizing that both $16^{\frac{1}{12}}$ and $\sqrt[12]{16}$ represent the same value, and thus, answers a and b are correct.