If the surds `root(4)(4), root(6)(5), root(8)(6) and root(12)(8)` are arranged in ascending order from left to right, then the third surd from the left is

Options

A. `root(12)(8)`
B. `root(4)(4)`
C. `root(8)(6)`
D. `root(6)(5)`

Explanation

To solve the problem of arranging the surds \( \sqrt[4]{4}, \sqrt[6]{5}, \sqrt[8]{6}, \sqrt[12]{8} \) in ascending order, we will follow these steps:

### Step 1: Rewrite the surds in exponential form
We can express each surd in terms of powers:
- \( \sqrt[4]{4} = 4^{1/4} \)
- \( \sqrt[6]{5} = 5^{1/6} \)
- \( \sqrt[8]{6} = 6^{1/8} \)
- \( \sqrt[12]{8} = 8^{1/12} \)

### Step 2: Convert the bases to powers of 2
Next, we will express the bases in terms of powers of 2:
- \( 4 = 2^2 \) so \( 4^{1/4} = (2^2)^{1/4} = 2^{2/4} = 2^{1/2} = \sqrt{2} \)
- \( 5 \) remains \( 5^{1/6} \)
- \( 6 = 2 \cdot 3 \) so \( 6^{1/8} = (2 \cdot 3)^{1/8} = 2^{1/8} \cdot 3^{1/8} \)
- \( 8 = 2^3 \) so \( 8^{1/12} = (2^3)^{1/12} = 2^{3/12} = 2^{1/4} \)

### Step 3: Compare the surds
Now we have:
- \( \sqrt{2} = 2^{1/2} \)
- \( 5^{1/6} \)
- \( 2^{1/8} \cdot 3^{1/8} \)
- \( 2^{1/4} \)

### Step 4: Finding a common exponent
To compare these surds, we can express them with a common exponent. The least common multiple of the denominators (4, 6, 8, 12) is 24. We will express each term with a power of \( 24 \):
- \( \sqrt{2} = 2^{1/2} = 2^{12/24} \)
- \( 5^{1/6} = 5^{4/24} \)
- \( 2^{1/8} \cdot 3^{1/8} = 2^{3/24} \cdot 3^{3/24} \)
- \( 2^{1/4} = 2^{6/24} \)

### Step 5: Rewrite the surds
Now we can rewrite the surds:
- \( 2^{12/24} \)
- \( 5^{4/24} \)
- \( 2^{3/24} \cdot 3^{3/24} \)
- \( 2^{6/24} \)

### Step 6: Calculate the values
We need to calculate the approximate values of these surds:
- \( 2^{12/24} = 2^{0.5} \approx 1.414 \)
- \( 5^{4/24} = 5^{0.1667} \approx 1.379 \)
- \( 2^{3/24} \cdot 3^{3/24} = (2^{0.125} \cdot 3^{0.125}) \approx 1.1447 \)
- \( 2^{6/24} = 2^{0.25} \approx 1.189 \)

### Step 7: Arrange in ascending order
Now we can arrange these values in ascending order:
1. \( 2^{3/24} \cdot 3^{3/24} \approx 1.1447 \)
2. \( 2^{6/24} \approx 1.189 \)
3. \( 5^{4/24} \approx 1.379 \)
4. \( 2^{12/24} \approx 1.414 \)

### Conclusion
The third surd from the left is \( \sqrt[6]{5} \).