State, whether the following numbers are rational or not:
`(sqrt(3)-sqrt(2))^(2)`

Answer

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

### Step 1: Expand the expression using the formula for the square of a binomial.
The formula for the square of a binomial is:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]
In our case, let \(a = \sqrt{3}\) and \(b = \sqrt{2}\). Therefore, we can write:
\[ (\sqrt{3} - \sqrt{2})^2 = (\sqrt{3})^2 - 2(\sqrt{3})(\sqrt{2}) + (\sqrt{2})^2 \]

### Step 2: Calculate each term in the expansion.
- \((\sqrt{3})^2 = 3\)
- \((\sqrt{2})^2 = 2\)
- \(2(\sqrt{3})(\sqrt{2}) = 2\sqrt{6}\)

### Step 3: Substitute the calculated values back into the expression.
Now substituting these values back, we get:
\[ (\sqrt{3} - \sqrt{2})^2 = 3 - 2\sqrt{6} + 2 \]

### Step 4: Simplify the expression.
Combine the constant terms:
\[ 3 + 2 - 2\sqrt{6} = 5 - 2\sqrt{6} \]

### Step 5: Determine if the result is rational or irrational.
The term \(5\) is rational, but \(2\sqrt{6}\) is irrational because \(\sqrt{6}\) is not a perfect square. The difference between a rational number (5) and an irrational number (2√6) is always irrational.

### Conclusion:
Thus, \((\sqrt{3} - \sqrt{2})^2 = 5 - 2\sqrt{6}\) is an **irrational number**.

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