Solve Nested Radicals: Cube Root of Square Root of 144

To solve this problem, let's follow these steps:

• Express the square root as a fractional exponent.
• Express the cube root as another fractional exponent.
• Multiply the exponents together using the rule $(a^m)^n = a^{m \times n}$.
• Recapture the result as a root expression.

Let's apply these steps:
Step 1: The square root of 144 can be expressed as $144^{1/2}$.
Step 2: We need the cube root of this expression, so we have $(144^{1/2})^{1/3}$.
Step 3: Using the property of exponents $(a^m)^n = a^{m \times n}$, we multiply the exponents: $(144^{1/2})^{1/3} = 144^{(1/2) \times (1/3)} = 144^{1/6}$.
Step 4: Re-express this as a root: Since $144^{1/6}$ is equivalent to the sixth root, we have $\sqrt[6]{144}$.

Therefore, the solution to the problem is $\sqrt[6]{144}$, which corresponds to choice 3.