The smallest among `root6(12)`, `root3(4)`, `root4(5)`, `sqrt(3)` is

Options

A. A) `root6(12)`
B. B) `root3(4)`
C. C) `sqrt(3)`
D. D) `root4(5)`

Answer

To find the smallest among \( \sqrt[6]{12} \), \( \sqrt[3]{4} \), \( \sqrt[4]{5} \), and \( \sqrt{3} \), we can express each root in terms of powers with a common denominator. Here’s how to do it step by step:

### Step 1: Express each root in exponent form
1. **6th root of 12**:
\[ \sqrt[6]{12} = 12^{1/6} \]
2. **Cube root of 4**:
\[ \sqrt[3]{4} = 4^{1/3} \]
3. **4th root of 5**:
\[ \sqrt[4]{5} = 5^{1/4} \]
4. **Square root of 3**:
\[ \sqrt{3} = 3^{1/2} \]

### Step 2: Find a common denominator for the exponents
The denominators of the exponents are 6, 3, 4, and 2. The least common multiple (LCM) of these numbers is 12.

### Step 3: Rewrite each expression with the common denominator
1. **6th root of 12**:
\[ 12^{1/6} = 12^{2/12} \]
2. **Cube root of 4**:
\[ 4^{1/3} = 4^{4/12} \]
3. **4th root of 5**:
\[ 5^{1/4} = 5^{3/12} \]
4. **Square root of 3**:
\[ 3^{1/2} = 3^{6/12} \]

### Step 4: Convert bases to powers of 12
Now we need to express all these bases in terms of powers of 12 to compare them easily.

1. **Convert \( 12^{2/12} \)**:
\[ 12^{2/12} = 12^{1/6} \]
2. **Convert \( 4^{4/12} \)**:
\[ 4 = 2^2 \implies 4^{4/12} = (2^2)^{4/12} = 2^{8/12} = 2^{2/3} \]
3. **Convert \( 5^{3/12} \)**:
\[ 5^{3/12} = 5^{1/4} \]
4. **Convert \( 3^{6/12} \)**:
\[ 3^{6/12} = 3^{1/2} \]

### Step 5: Calculate approximate values
Now we can calculate the approximate values for comparison:
1. **Calculate \( 12^{1/6} \)**:
\[ 12^{1/6} \approx 1.348 \]
2. **Calculate \( 4^{1/3} \)**:
\[ 4^{1/3} \approx 1.587 \]
3. **Calculate \( 5^{1/4} \)**:
\[ 5^{1/4} \approx 1.495 \]
4. **Calculate \( 3^{1/2} \)**:
\[ 3^{1/2} \approx 1.732 \]

### Step 6: Compare the values
Now we can compare the approximate values:
- \( 12^{1/6} \approx 1.348 \)
- \( 4^{1/3} \approx 1.587 \)
- \( 5^{1/4} \approx 1.495 \)
- \( 3^{1/2} \approx 1.732 \)

### Conclusion
The smallest value among these is \( 12^{1/6} \) or \( \sqrt[6]{12} \).

### Final Answer
Thus, the smallest among \( \sqrt[6]{12} \), \( \sqrt[3]{4} \), \( \sqrt[4]{5} \), and \( \sqrt{3} \) is:
\[ \sqrt[6]{12} \]