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

Answer

To determine whether the number \(\left(\frac{3}{2\sqrt{2}}\right)^{2}\) is rational or not, we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \left(\frac{3}{2\sqrt{2}}\right)^{2} \]

### Step 2: Apply the property of squares
Using the property that \(\left(\frac{a}{b}\right)^{2} = \frac{a^{2}}{b^{2}}\), we can rewrite the expression:
\[ \left(\frac{3}{2\sqrt{2}}\right)^{2} = \frac{3^{2}}{(2\sqrt{2})^{2}} \]

### Step 3: Calculate the numerator
Now, calculate the square of the numerator:
\[ 3^{2} = 9 \]

### Step 4: Calculate the denominator
Next, we calculate the square of the denominator:
\[ (2\sqrt{2})^{2} = 2^{2} \cdot (\sqrt{2})^{2} = 4 \cdot 2 = 8 \]

### Step 5: Combine the results
Now we can combine the results from the numerator and the denominator:
\[ \frac{3^{2}}{(2\sqrt{2})^{2}} = \frac{9}{8} \]

### Step 6: Determine if the result is rational
The fraction \(\frac{9}{8}\) is a ratio of two integers (9 and 8), which means it is a rational number.

### Conclusion
Thus, we conclude that:
\[ \left(\frac{3}{2\sqrt{2}}\right)^{2} \text{ is a rational number.} \]

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