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

Video Solution

Solution Steps

00:00 Simplify the following problem

00:03 Every number is to the power of 1

00:09 When we have a root of the order (B) on a number (X) to the power of (A)

00:14 The result equals number (X) to the power of (A divided by B)

00:19 Apply this formula to our exercise

00:26 When we have division of powers (A/B) with equal bases

00:31 The result equals the common base to the power of the difference of exponents (A - B)

00:36 Apply this formula to our exercise, and subtract between the powers

00:43 Determine the common denominator and proceed to calculate the power

00:50 This is the solution

Step-by-Step Solution

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 $3^{\frac{1}{12}}$ .