which one of the following is the least ?
`sqrt(3),root3(2),sqrt(2)` and `root3(4)`

Options

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

Answer

To find which of the following numbers is the least among \( \sqrt{3}, \sqrt[3]{2}, \sqrt{2}, \text{ and } \sqrt[3]{4} \), we can compare them by expressing each in terms of a common exponent.

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

### Step 2: Find a common denominator for the exponents
The exponents we have are \( \frac{1}{2}, \frac{1}{3}, \frac{1}{2}, \text{ and } \frac{2}{3} \). The least common multiple (LCM) of the denominators (2 and 3) is 6.

### Step 3: Convert each expression to have the same denominator
- For \( \sqrt{3} = 3^{1/2} \):
\[ 3^{1/2} = 3^{3/6} \]
- For \( \sqrt[3]{2} = 2^{1/3} \):
\[ 2^{1/3} = 2^{2/6} \]
- For \( \sqrt{2} = 2^{1/2} \):
\[ 2^{1/2} = 2^{3/6} \]
- For \( \sqrt[3]{4} = 2^{2/3} \):
\[ 2^{2/3} = 2^{4/6} \]

### Step 4: Compare the bases raised to the same power
Now we can compare:
- \( 3^{3/6} \)
- \( 2^{2/6} \)
- \( 2^{3/6} \)
- \( 2^{4/6} \)

### Step 5: Calculate the values
Now we can evaluate the bases raised to the power \( \frac{1}{6} \):
1. \( 3^{3/6} = 3^{0.5} = \sqrt{3} \approx 1.732 \)
2. \( 2^{2/6} = 2^{0.333} \approx 1.260 \)
3. \( 2^{3/6} = 2^{0.5} = \sqrt{2} \approx 1.414 \)
4. \( 2^{4/6} = 2^{0.667} \approx 1.587 \)

### Step 6: Identify the least value
From the approximations:
- \( \sqrt{3} \approx 1.732 \)
- \( \sqrt[3]{2} \approx 1.260 \)
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt[3]{4} \approx 1.587 \)

The least value is \( \sqrt[3]{2} \).

### Conclusion
Thus, the least among \( \sqrt{3}, \sqrt[3]{2}, \sqrt{2}, \text{ and } \sqrt[3]{4} \) is \( \sqrt[3]{2} \).

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