Examine whether the following numbers are rational or irrational :
` (2 sqrt3 - 3 sqrt2) (2sqrt3 + 3 sqrt2)`

Answer

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

### Step 1: Recognize the Expression
The expression is in the form of \( (a - b)(a + b) \), which is a difference of squares. Here, \( a = 2\sqrt{3} \) and \( b = 3\sqrt{2} \).

### Step 2: Apply the Difference of Squares Formula
Using the formula \( (a - b)(a + b) = a^2 - b^2 \):

\[ (2\sqrt{3})^2 - (3\sqrt{2})^2 \]

### Step 3: Calculate \( a^2 \)
Calculate \( (2\sqrt{3})^2 \):

\[ (2\sqrt{3})^2 = 2^2 \cdot (\sqrt{3})^2 = 4 \cdot 3 = 12 \]

### Step 4: Calculate \( b^2 \)
Calculate \( (3\sqrt{2})^2 \):

\[ (3\sqrt{2})^2 = 3^2 \cdot (\sqrt{2})^2 = 9 \cdot 2 = 18 \]

### Step 5: Substitute Back into the Expression
Now substitute \( a^2 \) and \( b^2 \) back into the difference of squares formula:

\[ 12 - 18 \]

### Step 6: Simplify the Expression
Now simplify:

\[ 12 - 18 = -6 \]

### Conclusion
Since \(-6\) is a whole number, it is a rational number.

Thus, the expression \( (2\sqrt{3} - 3\sqrt{2})(2\sqrt{3} + 3\sqrt{2}) \) is rational.

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