`sqrt7-3-sqrt2` is:

Options

A. a rational number
B. a natural number
C. equal to zero
D. an irrational number

Answer

To determine whether the expression \( \sqrt{7} - 3 - \sqrt{2} \) is a rational number, a natural number, or an irrational number, we can follow these steps:

### Step 1: Approximate the values of the square roots
First, we need to find the approximate values of \( \sqrt{7} \) and \( \sqrt{2} \).

- \( \sqrt{7} \) is approximately \( 2.64575 \)
- \( \sqrt{2} \) is approximately \( 1.41421 \)

### Step 2: Substitute the approximate values into the expression
Now, we can substitute these approximate values into the expression:

\[ \sqrt{7} - 3 - \sqrt{2} \approx 2.64575 - 3 - 1.41421 \]

### Step 3: Perform the subtraction
Now, we perform the subtraction step-by-step:

1. First, calculate \( 2.64575 - 3 \):
\[ 2.64575 - 3 = -0.35425 \]

2. Next, subtract \( \sqrt{2} \):
\[ -0.35425 - 1.41421 = -1.76846 \]

### Step 4: Analyze the result
The result of the expression \( \sqrt{7} - 3 - \sqrt{2} \) is approximately \( -1.76846 \).

### Step 5: Determine the type of number
1. **Check if it equals 0**: The result is not equal to 0.
2. **Check if it is a natural number**: Natural numbers are positive integers (1, 2, 3, ...). Since our result is negative, it is not a natural number.
3. **Check if it is a rational number**: A rational number can be expressed as a fraction of two integers. However, since both \( \sqrt{7} \) and \( \sqrt{2} \) are irrational numbers, their combination in this form results in an irrational number.

### Conclusion
Thus, the expression \( \sqrt{7} - 3 - \sqrt{2} \) is an **irrational number**.

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