The greatest of
`sqrt(2)`, `root6(3)`, `root3(4)`, `root4(5)` is

Options

A. `sqrt(2)`
B. `root6(3)`
C. `root3(4)`
D. `root4(5)`

Answer

To find the greatest among the numbers \( \sqrt{2} \), \( \sqrt[6]{3} \), \( \sqrt[3]{4} \), and \( \sqrt[4]{5} \), we will express each number in terms of powers with a common denominator.

### Step 1: Rewrite each expression with fractional exponents
- \( \sqrt{2} = 2^{1/2} \)
- \( \sqrt[6]{3} = 3^{1/6} \)
- \( \sqrt[3]{4} = 4^{1/3} \)
- \( \sqrt[4]{5} = 5^{1/4} \)

### Step 2: Find a common denominator for the exponents
The least common multiple (LCM) of the denominators \( 2, 6, 3, \) and \( 4 \) is \( 12 \).

### Step 3: Rewrite each expression using the common denominator
Now we will express each number with a power of \( 12 \):
- \( 2^{1/2} = 2^{6/12} \)
- \( 3^{1/6} = 3^{2/12} \)
- \( 4^{1/3} = 4^{4/12} \)
- \( 5^{1/4} = 5^{3/12} \)

### Step 4: Convert the bases to powers of \( 12 \)
Now we can express each number as follows:
- \( 2^{6/12} = 2^6 \)
- \( 3^{2/12} = 3^2 \)
- \( 4^{4/12} = 4^4 \)
- \( 5^{3/12} = 5^3 \)

### Step 5: Calculate the values of these powers
Now we compute the values:
- \( 2^6 = 64 \)
- \( 3^2 = 9 \)
- \( 4^4 = 256 \)
- \( 5^3 = 125 \)

### Step 6: Compare the results
Now we compare the calculated values:
- \( 64 \) (from \( \sqrt{2} \))
- \( 9 \) (from \( \sqrt[6]{3} \))
- \( 256 \) (from \( \sqrt[3]{4} \))
- \( 125 \) (from \( \sqrt[4]{5} \))

### Step 7: Identify the greatest value
The greatest value among \( 64, 9, 256, \) and \( 125 \) is \( 256 \).

### Conclusion
Thus, the greatest of \( \sqrt{2} \), \( \sqrt[6]{3} \), \( \sqrt[3]{4} \), and \( \sqrt[4]{5} \) is \( \sqrt[3]{4} \).