For the expression `xsqrtx` , where -100 `le` x `le ` 100 , how many x values are there such that the expression is an integer ?

Options

A. 100
B. 21
C. 20
D. 11

Answer

To solve the problem of finding how many integer values of \( x \) satisfy the expression \( x\sqrt{x} \) for \( -100 \leq x \leq 100 \), we can follow these steps:

### Step 1: Understand the expression
The expression \( x\sqrt{x} \) involves the square root of \( x \). For \( \sqrt{x} \) to be defined and real, \( x \) must be non-negative. Therefore, we restrict our consideration to \( 0 \leq x \leq 100 \).

**Hint:** Remember that the square root of a negative number is not a real number.

### Step 2: Identify conditions for \( \sqrt{x} \) to be an integer
For \( \sqrt{x} \) to be an integer, \( x \) must be a perfect square. This means \( x \) can take values like \( 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 \).

**Hint:** A perfect square is an integer that is the square of an integer.

### Step 3: List the perfect squares in the range
The perfect squares from \( 0 \) to \( 100 \) are:
- \( 0^2 = 0 \)
- \( 1^2 = 1 \)
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)
- \( 6^2 = 36 \)
- \( 7^2 = 49 \)
- \( 8^2 = 64 \)
- \( 9^2 = 81 \)
- \( 10^2 = 100 \)

**Hint:** Count the integers from \( 0 \) to \( 10 \) to find the perfect squares.

### Step 4: Count the perfect squares
The perfect squares listed are:
- \( 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 \)

There are a total of 11 perfect squares in this range.

**Hint:** Make sure to include \( 0 \) as a perfect square.

### Conclusion
Thus, the number of integer values of \( x \) such that \( x\sqrt{x} \) is an integer is **11**.

**Final Answer:** The correct answer is **11**.