If `A = sqrt(2) , B = root(3)(3) and C = root(4)(4)` then which of the relation is correct ?

Options

A. A > C
B. A > B = C
C. A = C >B
D. None of these

Answer

To solve the problem, we need to evaluate the values of A, B, and C and then compare them to determine the correct relation among them.

### Step 1: Calculate the value of A
Given:
\[ A = \sqrt{2} \]

Calculating \( A \):
\[ A \approx 1.414 \]

### Step 2: Calculate the value of B
Given:
\[ B = \sqrt[3]{3} \]

Calculating \( B \):
To find \( \sqrt[3]{3} \), we can use a calculator or estimate:
\[ B \approx 1.442 \]

### Step 3: Calculate the value of C
Given:
\[ C = \sqrt[4]{4} \]

Calculating \( C \):
Since \( \sqrt[4]{4} = \sqrt{2} \):
\[ C = \sqrt{2} \approx 1.414 \]

### Step 4: Compare the values of A, B, and C
Now we have:
- \( A \approx 1.414 \)
- \( B \approx 1.442 \)
- \( C \approx 1.414 \)

From the values calculated:
- \( A = C \)
- \( A < B \)

### Conclusion
Thus, the correct relation is:
\[ A = C < B \]