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

Video Solution

Solution Steps

00:00 Simplify the following problem

00:03 When we have a root of the order (B) on a number (X) to the power of (A)

00:07 The result equals the number (X) to the power of (A divided by B)

00:10 Any number is to the power of 1

00:17 Apply this formula to our exercise

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

00:33 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

00:41 Determine the common denominator and calculate the power

00:44 This is the solution

Step-by-Step Solution

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

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