Simplify the Fraction: Fourth Root of 3 Divided by Sixth Root of 3

To solve this problem, we must simplify the expression $\frac{\sqrt[4]{3}}{\sqrt[6]{3}}$. We will follow these steps:

• Step 1: Convert roots to exponents.
  $\sqrt[4]{3}$ is equivalent to $3^{\frac{1}{4}}$, and $\sqrt[6]{3}$ is equivalent to $3^{\frac{1}{6}}$.
• Step 2: Apply the quotient of powers formula.
  Using the property $\frac{a^m}{a^n} = a^{m-n}$, we have: $\frac{3^{\frac{1}{4}}}{3^{\frac{1}{6}}} = 3^{\frac{1}{4} - \frac{1}{6}}$
• Step 3: Perform the subtraction of the exponents.
  To subtract the fractions $\frac{1}{4}$ and $\frac{1}{6}$, find a common denominator. The least common multiple of 4 and 6 is 12, so: $\frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}$ Thus, $\frac{1}{4} - \frac{1}{6} = \frac{3}{12} - \frac{2}{12} = \frac{1}{12}$
• Step 4: Simplify the result.
  Thus, we have: $3^{\frac{1}{4} - \frac{1}{6}} = 3^{\frac{1}{12}}$

Hence, the simplified form of the given expression is $3^{\frac{1}{12}}$.