`sqrt(X)` is a perfect square. Which of the following is/are true?

Options

A. X is a perfect square.
B. `X^(2)` is a perfect square
C. Neighte a nor b
D. Both a and b

Answer

To determine which of the given statements are true based on the condition that \( \sqrt{X} \) is a perfect square, let's analyze the problem step by step.

### Step 1: Understanding the Condition
We know that \( \sqrt{X} \) is a perfect square. This means that there exists some integer \( n \) such that:
\[ \sqrt{X} = n^2 \]
Squaring both sides gives:
\[ X = (n^2)^2 = n^4 \]
This implies that \( X \) is also a perfect square since \( n^4 \) can be expressed as \( (n^2)^2 \).

### Step 2: Analyzing Statement 1
The first statement is: "Root \( X \) is a perfect square."
Since we have established that \( \sqrt{X} = n^2 \), and \( n^2 \) is indeed a perfect square (as it is the square of an integer), this statement is true.

### Step 3: Analyzing Statement 2
The second statement is: "X square is a perfect square."
We need to check if \( X^2 \) is a perfect square. Since we have \( X = n^4 \), we can find \( X^2 \):
\[ X^2 = (n^4)^2 = n^8 \]
Since \( n^8 \) is also a perfect square (as it can be expressed as \( (n^4)^2 \)), this statement is also true.

### Step 4: Conclusion
Both statements are true. Therefore, the correct answer is that both statements A and B are true.

### Final Answer
Both statements are true.