Solve: Division of Cube Root and Fourth Root of 5

Video Solution

Solution Steps

00:00 Simplify the following problem

00:03 Every number is to the power of 1

00:11 When we have a root of the order (B) on a number (X) to the power of (A)

00:15 The result equals number (X) to the power of (A divided by B)

00:20 Apply this formula to our exercise

00:27 When we have division of powers (A/B) with equal bases

00:36 The result equals the common base to the power of the difference of powers (A - B)

00:39 Apply this formula to our exercise, and subtract between the powers

00:49 Identify the common denominator and calculate the power

00:58 Apply the formula again and convert the power to the 12th root

01:05 This is the solution

Step-by-Step Solution

To solve the given problem, let us proceed step by step:

Step 1: Convert the given roots into fractional exponents:
The cube root $\sqrt[3]{5}$ can be expressed as $5^{1/3}$ .
The fourth root $\sqrt[4]{5}$ can be expressed as $5^{1/4}$ .
Step 2: Apply the quotient rule for exponents:
For division, $\frac{a^m}{a^n} = a^{m-n}$ .
Thus, $\frac{5^{1/3}}{5^{1/4}} = 5^{1/3 - 1/4}$ .
Step 3: Perform the subtraction of the exponents:
$1/3 - 1/4 = \frac{4}{12} - \frac{3}{12} = \frac{1}{12}$ .
Therefore, we have $5^{\frac{1}{12}}$ .
Step 4: Compare with answer choices:
- The expression $5^{\frac{1}{12}}$ directly matches Choice 3.
- Alternatively, we can express this as a 12th root: $\sqrt[12]{5}$ , which matches Choice 1.

Both answer choices (a) $\sqrt[12]{5}$ and (c) $5^{\frac{1}{12}}$ correctly represent the simplified form of the expression.

Thus, the correct solution to the problem is given by Answers (a) and (c).