Find the product of `sqrt5,root(6)(6)androot(3)(4)`.

Answer

To find the product of \( \sqrt{5} \), \( 6\sqrt[6]{6} \), and \( \sqrt[3]{4} \), we will follow these steps:

### Step 1: Rewrite each term in exponential form
- \( \sqrt{5} = 5^{1/2} \)
- \( 6\sqrt[6]{6} = 6^{1/6} \)
- \( \sqrt[3]{4} = 4^{1/3} \)

### Step 2: Combine the terms
We need to find the product:
\[ \sqrt{5} \times 6\sqrt[6]{6} \times \sqrt[3]{4} = 5^{1/2} \times 6^{1/6} \times 4^{1/3} \]

### Step 3: Simplify the expression
We can rewrite \( 4 \) as \( 2^2 \):
\[ 4^{1/3} = (2^2)^{1/3} = 2^{2/3} \]
Now we have:
\[ 5^{1/2} \times 6^{1/6} \times 2^{2/3} \]

### Step 4: Rewrite \( 6 \) in terms of its prime factors
Since \( 6 = 2 \times 3 \), we can express it as:
\[ 6^{1/6} = (2 \times 3)^{1/6} = 2^{1/6} \times 3^{1/6} \]

### Step 5: Combine all terms
Now we can combine all the terms:
\[ 5^{1/2} \times 2^{2/3} \times 2^{1/6} \times 3^{1/6} \]
Combine the powers of \( 2 \):
\[ 2^{2/3 + 1/6} \]
To add the exponents, we need a common denominator. The common denominator of \( 3 \) and \( 6 \) is \( 6 \):
\[ 2/3 = 4/6 \quad \text{so} \quad 2^{4/6 + 1/6} = 2^{5/6} \]

### Step 6: Write the final expression
Now we can write the entire product:
\[ = 5^{1/2} \times 2^{5/6} \times 3^{1/6} \]

### Step 7: Final Answer
Thus, the product of \( \sqrt{5} \), \( 6\sqrt[6]{6} \), and \( \sqrt[3]{4} \) is:
\[ 5^{1/2} \times 2^{5/6} \times 3^{1/6} \]