Which among the following numbers is the greatest?

Options

A. `root(6)(16)`
B. `root(3)(4)`
C. `root(4)(5)`
D. `sqrt(2)`

Answer

To determine which among the following numbers is the greatest, we will evaluate the four given expressions step by step.

### Step 1: Evaluate the 6th root of 16
We start with the first expression, which is the 6th root of 16.

\[ \sqrt[6]{16} = \sqrt[6]{4^2} \]

Now, we can simplify this:

\[ \sqrt[6]{4^2} = \sqrt[6]{4^{2}} = 4^{\frac{2}{6}} = 4^{\frac{1}{3}} = \sqrt[3]{4} \]

Next, we calculate the value of \(\sqrt[3]{4}\):

\[ \sqrt[3]{4} \approx 1.5874 \]

### Step 2: Evaluate the cube root of 4
The second expression is the cube root of 4:

\[ \sqrt[3]{4} \approx 1.5874 \]

### Step 3: Evaluate \(4^{\sqrt{5}}\)
The third expression is \(4^{\sqrt{5}}\). We can rewrite this as:

\[ 4^{\sqrt{5}} = (2^2)^{\sqrt{5}} = 2^{2\sqrt{5}} \]

Now we need to calculate \(2^{2\sqrt{5}}\). First, we find \(\sqrt{5}\):

\[ \sqrt{5} \approx 2.2360 \]

Thus,

\[ 2^{2\sqrt{5}} = 2^{2 \times 2.2360} = 2^{4.472} \approx 21.086 \]

### Step 4: Evaluate \(\sqrt{4}\)
The fourth expression is simply the square root of 4:

\[ \sqrt{4} = 2 \]

### Step 5: Compare the values
Now we have the approximate values of all four expressions:

1. \(\sqrt[6]{16} \approx 1.5874\)
2. \(\sqrt[3]{4} \approx 1.5874\)
3. \(4^{\sqrt{5}} \approx 21.086\)
4. \(\sqrt{4} = 2\)

### Conclusion
Among these values, the greatest is \(4^{\sqrt{5}} \approx 21.086\).

### Final Answer
The greatest number among the given options is \(4^{\sqrt{5}}\).

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