Simplify the Expression: Cube Root of 4 Divided by Sixth Root of 4

To solve this problem, we will convert the roots into exponent form and simplify:

• Step 1: Express each root as an exponent.

• Step 2: Simplify the expression using the properties of exponents.

Let's apply each step:

Step 1: Convert $\sqrt[3]{4}$ and $\sqrt[6]{4}$ into exponential form:

$\sqrt[3]{4} = 4^{\frac{1}{3}} \quad \text{and} \quad \sqrt[6]{4} = 4^{\frac{1}{6}}$

Step 2: Apply the quotient rule for exponents $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{4^{\frac{1}{3}}}{4^{\frac{1}{6}}} = 4^{\frac{1}{3} - \frac{1}{6}}$

We need to subtract the exponents. First, find a common denominator for the fractions:

$\frac{1}{3} - \frac{1}{6} = \frac{2}{6} - \frac{1}{6} = \frac{1}{6}$

The expression simplifies to:

$4^{\frac{1}{6}}$

Therefore, the simplified result is $4^\frac{1}{6}$.

Comparing this with the answer choices, we conclude that the correct choice is $4^{\frac{1}{6}}$.