State, whether the following numbers are rational or not:
`((sqrt(7))/(6sqrt(2)))^(2)`

Answer

To determine whether the number \(\left(\frac{\sqrt{7}}{6\sqrt{2}}\right)^{2}\) is rational or not, we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \left(\frac{\sqrt{7}}{6\sqrt{2}}\right)^{2} \]

### Step 2: Apply the property of squares
Using the property of squares, we can rewrite the expression as:
\[ \frac{(\sqrt{7})^{2}}{(6\sqrt{2})^{2}} \]

### Step 3: Simplify the numerator and denominator
Now, we simplify the numerator and denominator:
- The numerator: \((\sqrt{7})^{2} = 7\)
- The denominator: \((6\sqrt{2})^{2} = 6^{2} \cdot (\sqrt{2})^{2} = 36 \cdot 2 = 72\)

So, we have:
\[ \frac{7}{72} \]

### Step 4: Determine if the result is rational
The fraction \(\frac{7}{72}\) is in the form \(\frac{p}{q}\) where \(p = 7\) and \(q = 72\). Since both 7 and 72 are integers and 72 is not zero, this fraction is a rational number.

### Conclusion
Thus, we conclude that \(\left(\frac{\sqrt{7}}{6\sqrt{2}}\right)^{2}\) is a rational number.

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