The least one of `2sqrt(3)`, `2root4(5)`, `sqrt(8)` and `3sqrt(2)` is

Options

A. A) `2sqrt(3)`
B. B) `2root4(5)`
C. C) `sqrt(8)`
D. D) `3sqrt(2)`

Answer

To find the least one among the numbers \(2\sqrt{3}\), \(2\sqrt[4]{5}\), \(\sqrt{8}\), and \(3\sqrt{2}\), we will convert all of them into a common root form, specifically the fourth root, for easier comparison.

### Step-by-step Solution:

1. **Convert \(2\sqrt{3}\) to fourth root form**:
\[ 2\sqrt{3} = 2 \cdot 3^{1/2} = 2^{2/2} \cdot 3^{1/2} = \sqrt{2^2 \cdot 3} = \sqrt{4 \cdot 3} = \sqrt{12} \]
To convert to fourth root:
\[ \sqrt{12} = \sqrt{(12)^{2/4}} = \sqrt{144} = \sqrt{12} \]

2. **Convert \(2\sqrt[4]{5}\) to fourth root form**:
\[ 2\sqrt[4]{5} = 2 \cdot 5^{1/4} = \sqrt[4]{2^4 \cdot 5} = \sqrt[4]{16 \cdot 5} = \sqrt[4]{80} \]

3. **Convert \(\sqrt{8}\) to fourth root form**:
\[ \sqrt{8} = 8^{1/2} = (2^3)^{1/2} = 2^{3/2} = 2^{6/4} = \sqrt[4]{2^6} = \sqrt[4]{64} \]

4. **Convert \(3\sqrt{2}\) to fourth root form**:
\[ 3\sqrt{2} = 3 \cdot 2^{1/2} = 3^{2/2} \cdot 2^{1/2} = \sqrt{3^2 \cdot 2} = \sqrt{9 \cdot 2} = \sqrt{18} \]
To convert to fourth root:
\[ \sqrt{18} = \sqrt{(18)^{2/4}} = \sqrt{324} = \sqrt{18} \]

5. **Now we have all numbers in fourth root form**:
- \(2\sqrt{3} = \sqrt[4]{144}\)
- \(2\sqrt[4]{5} = \sqrt[4]{80}\)
- \(\sqrt{8} = \sqrt[4]{64}\)
- \(3\sqrt{2} = \sqrt[4]{324}\)

6. **Compare the values**:
- \(\sqrt[4]{144} \approx 3.74\)
- \(\sqrt[4]{80} \approx 3.17\)
- \(\sqrt[4]{64} = 2\)
- \(\sqrt[4]{324} \approx 4.64\)

7. **Identify the least value**:
The smallest value among these is \(\sqrt[4]{64}\).

### Conclusion:
The least one of \(2\sqrt{3}\), \(2\sqrt[4]{5}\), \(\sqrt{8}\), and \(3\sqrt{2}\) is \(\sqrt{8}\) or \(2\sqrt{2}\).