Which one is greatest out of `sqrt2,root(6)(3),root(3)(4)androot(4)(5)`?

Options

A. `root(3)(4)`
B. `root(4)(5)`
C. `sqrt2`
D. `root(6)(3)`

Answer

To determine which of the numbers \( \sqrt{2}, \sqrt[6]{3}, \sqrt[3]{4}, \) and \( \sqrt[4]{5} \) is the greatest, we can express each number in terms of a common base using indices.

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

### Step 2: Find a common denominator for the exponents
To compare these numbers, we need to express them with a common denominator. The denominators of the exponents are 2, 6, 3, and 4. The least common multiple (LCM) of these numbers is 12.

### Step 3: Rewrite each expression with the common denominator
- For \( 2^{1/2} \):
\[ 2^{1/2} = 2^{6/12} \]
- For \( 3^{1/6} \):
\[ 3^{1/6} = 3^{2/12} \]
- For \( 2^{2/3} \):
\[ 2^{2/3} = 2^{8/12} \]
- For \( 5^{1/4} \):
\[ 5^{1/4} = 5^{3/12} \]

### Step 4: Compare the expressions
Now we can compare:
- \( 2^{6/12} \)
- \( 3^{2/12} \)
- \( 2^{8/12} \)
- \( 5^{3/12} \)

### Step 5: Calculate the values of the bases raised to the respective powers
- \( 2^{6/12} = 2^{0.5} = \sqrt{2} \approx 1.414 \)
- \( 3^{2/12} = 3^{1/6} \approx 1.200 \)
- \( 2^{8/12} = 2^{0.6667} \approx 1.587 \)
- \( 5^{3/12} = 5^{0.25} \approx 1.495 \)

### Step 6: Determine the greatest value
Now we can see:
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt[6]{3} \approx 1.200 \)
- \( \sqrt[3]{4} \approx 1.587 \)
- \( \sqrt[4]{5} \approx 1.495 \)

The greatest value among these is \( \sqrt[3]{4} \).

### Final Answer
Thus, the greatest number among \( \sqrt{2}, \sqrt[6]{3}, \sqrt[3]{4}, \) and \( \sqrt[4]{5} \) is \( \sqrt[3]{4} \).

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