Among the numbers `sqrt(2)`, `root3(9)`, `root4(16)`, `root5(32)`, the greatest one is

Options

A. `sqrt(2)`
B. `root3(9)`
C. `root4(16)`
D. `root5(32)`

Answer

To determine which of the numbers \( \sqrt{2} \), \( \sqrt[3]{9} \), \( \sqrt[4]{16} \), and \( \sqrt[5]{32} \) is the greatest, we can express each of these numbers in terms of powers and then compare them.

### Step 1: Rewrite each expression in exponent form
1. \( \sqrt{2} = 2^{1/2} \)
2. \( \sqrt[3]{9} = 9^{1/3} = (3^2)^{1/3} = 3^{2/3} \)
3. \( \sqrt[4]{16} = 16^{1/4} = (2^4)^{1/4} = 2^{4/4} = 2^1 = 2 \)
4. \( \sqrt[5]{32} = 32^{1/5} = (2^5)^{1/5} = 2^{5/5} = 2^1 = 2 \)

### Step 2: Find a common exponent to compare the numbers
To compare these numbers, we can raise each number to a common power. A good choice is to raise them to the 6th power (the least common multiple of the denominators of the exponents).

1. \( (2^{1/2})^6 = 2^{3} = 8 \)
2. \( (3^{2/3})^6 = 3^{4} = 81 \)
3. \( (2^1)^6 = 2^{6} = 64 \)
4. \( (2^1)^6 = 2^{6} = 64 \)

### Step 3: Compare the results
Now we have:
- \( \sqrt{2}^6 = 8 \)
- \( \sqrt[3]{9}^6 = 81 \)
- \( \sqrt[4]{16}^6 = 64 \)
- \( \sqrt[5]{32}^6 = 64 \)

### Step 4: Identify the greatest value
From the calculations:
- \( 8 < 64 < 81 \)

Thus, the greatest number among \( \sqrt{2} \), \( \sqrt[3]{9} \), \( \sqrt[4]{16} \), and \( \sqrt[5]{32} \) is \( \sqrt[3]{9} \).

### Final Answer
The greatest number is \( \sqrt[3]{9} \).

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