Solve: Cube Root of 6² Divided by Fourth Root of 6²

To solve this problem, we'll follow these steps:

• Step 1: Convert the roots to fractional exponents.
• Step 2: Simplify the fractional exponents using properties of exponents.
• Step 3: Determine the simplified expression and verify the answer choice.

Now, let's work through each step:

Step 1: Convert the roots to fractional exponents.
The cube root $\sqrt[3]{6^2}$ can be expressed as $(6^2)^{1/3} = 6^{2/3}$.

The fourth root $\sqrt[4]{6^2}$ can be written as $(6^2)^{1/4} = 6^{2/4} = 6^{1/2}$.

Step 2: Simplify the quotient of these fractional exponents.
We have $\frac{6^{2/3}}{6^{1/2}}$.

Using the property of exponents $\frac{a^m}{a^n} = a^{m-n}$, we get:

$6^{2/3 - 1/2}$.

Step 3: Calculate $2/3 - 1/2$.
To subtract these fractions, find a common denominator. The common denominator of 3 and 2 is 6.

- Convert $2/3$ to sixths: $\frac{2 \cdot 2}{3 \cdot 2} = \frac{4}{6}$

- Convert $1/2$ to sixths: $\frac{1 \cdot 3}{2 \cdot 3} = \frac{3}{6}$

Perform the subtraction: $\frac{4}{6} - \frac{3}{6} = \frac{1}{6}$.

The expression simplifies to $6^{1/6}$.

Therefore, the simplified answer is $\mathbf{6^{\frac{1}{6}}}$, which corresponds to the correct choice: 3.

The solution to the problem is $\mathbf{6^{\frac{1}{6}}}$.