The problem is to simplify the expression $5^{\frac{1}{4}} \times 5^{\frac{1}{3}} \div 5^{\frac{1}{12}}$.

1. Problem Description

The problem is to simplify the expression $5^{\frac{1}{4}} \times 5^{\frac{1}{3}} \div 5^{\frac{1}{12}}$.

2. Solution Steps

To solve this problem, we will use the properties of exponents.

When multiplying exponents with the same base, we add the exponents:

$a^m \times a^n = a^{m+n}$

When dividing exponents with the same base, we subtract the exponents:

$a^m \div a^n = a^{m-n}$

Applying these rules to the given expression:

$5^{\frac{1}{4}} \times 5^{\frac{1}{3}} \div 5^{\frac{1}{12}} = 5^{\frac{1}{4} + \frac{1}{3} - \frac{1}{12}}$

First, we need to find a common denominator for the fractions $\frac{1}{4}$, $\frac{1}{3}$, and $\frac{1}{12}$. The least common denominator (LCD) of 4, 3, and 12 is

1

2. We convert each fraction to have a denominator of 12:

$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$

$\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}$

$\frac{1}{12}$ remains as $\frac{1}{12}$.

So, we have:

$5^{\frac{3}{12} + \frac{4}{12} - \frac{1}{12}} = 5^{\frac{3+4-1}{12}} = 5^{\frac{6}{12}}$

Now we simplify the fraction $\frac{6}{12}$:

$\frac{6}{12} = \frac{1}{2}$

Therefore, the expression simplifies to:

$5^{\frac{1}{2}}$

3. Final Answer

$5^{\frac{1}{2}}$