Whicn of the following is correct ?
`A=sqrt2,B=root(3)(3),C=root(4)(4)`

Options

A. `AgtB=C`
B. `BgtAgtC`
C. `BgtA=C`
D. `CgtA=B`

Answer

To solve the problem, we need to compare the values of A, B, and C given as follows:

- \( A = \sqrt{2} \)
- \( B = \sqrt[3]{3} \)
- \( C = \sqrt[4]{4} \)

### Step 1: Express A, B, and C in exponent form
We can express the square root, cube root, and fourth root in exponent form:
- \( A = 2^{1/2} \)
- \( B = 3^{1/3} \)
- \( C = 4^{1/4} \)

### Step 2: Rewrite C in terms of base 2
Since \( 4 = 2^2 \), we can rewrite C:
- \( C = (2^2)^{1/4} = 2^{2/4} = 2^{1/2} \)

### Step 3: Compare A and C
Now we have:
- \( A = 2^{1/2} \)
- \( C = 2^{1/2} \)

From this, we see that:
- \( A = C \)

### Step 4: Compare B with A and C
Next, we need to compare \( B = 3^{1/3} \) with \( A = 2^{1/2} \) and \( C = 2^{1/2} \).

### Step 5: Find a common exponent to compare A and B
To compare \( 2^{1/2} \) and \( 3^{1/3} \), we can find a common exponent. The least common multiple of the denominators (2 and 3) is 6.

- Rewrite \( A \) and \( B \) with a common exponent:
- \( A = 2^{1/2} = 2^{3/6} \)
- \( B = 3^{1/3} = 3^{2/6} \)

### Step 6: Compare the values
Now we need to compare \( 2^{3/6} \) and \( 3^{2/6} \):
- \( 2^3 = 8 \)
- \( 3^2 = 9 \)

Since \( 8 < 9 \), we have:
- \( 2^{3/6} < 3^{2/6} \)
- Thus, \( A < B \)

### Final Comparison
Now we have:
- \( A = C \)
- \( A < B \)

So, we conclude:
- \( B > A = C \)

### Conclusion
The correct option is:
- **B is greater than A and C (B > A = C)**