State, whether the following numbers are rational or not :
(i) `(2+sqrt(2))^(2)" (ii) "(5+sqrt(5))(5-sqrt(5))`
(iii) `(sqrt(7)/(5sqrt(2)))^(2)`

Answer

To determine whether the given numbers are rational or irrational, we will analyze each expression step by step.

### Step 1: Analyze the first expression \((2 + \sqrt{2})^2\)

1. Use the formula for the square of a binomial:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
Here, \(a = 2\) and \(b = \sqrt{2}\).

2. Calculate \(a^2\):
\[ 2^2 = 4 \]

3. Calculate \(2ab\):
\[ 2 \cdot 2 \cdot \sqrt{2} = 4\sqrt{2} \]

4. Calculate \(b^2\):
\[ (\sqrt{2})^2 = 2 \]

5. Combine all parts:
\[ (2 + \sqrt{2})^2 = 4 + 4\sqrt{2} + 2 = 6 + 4\sqrt{2} \]

6. Since \(\sqrt{2}\) is an irrational number, \(4\sqrt{2}\) is also irrational. Therefore, \(6 + 4\sqrt{2}\) is irrational.

### Conclusion for (i):
\((2 + \sqrt{2})^2\) is **irrational**.

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### Step 2: Analyze the second expression \((5 + \sqrt{5})(5 - \sqrt{5})\)

1. Use the difference of squares formula:
\[ (a + b)(a - b) = a^2 - b^2 \]
Here, \(a = 5\) and \(b = \sqrt{5}\).

2. Calculate \(a^2\):
\[ 5^2 = 25 \]

3. Calculate \(b^2\):
\[ (\sqrt{5})^2 = 5 \]

4. Combine the results:
\[ (5 + \sqrt{5})(5 - \sqrt{5}) = 25 - 5 = 20 \]

5. Since 20 is an integer, it is a rational number.

### Conclusion for (ii):
\((5 + \sqrt{5})(5 - \sqrt{5})\) is **rational**.

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### Step 3: Analyze the third expression \(\left(\frac{\sqrt{7}}{5\sqrt{2}}\right)^2\)

1. Square the numerator and the denominator:
\[ \left(\frac{\sqrt{7}}{5\sqrt{2}}\right)^2 = \frac{(\sqrt{7})^2}{(5\sqrt{2})^2} \]

2. Calculate the numerator:
\[ (\sqrt{7})^2 = 7 \]

3. Calculate the denominator:
\[ (5\sqrt{2})^2 = 5^2 \cdot (\sqrt{2})^2 = 25 \cdot 2 = 50 \]

4. Combine the results:
\[ \frac{7}{50} \]

5. Since \(\frac{7}{50}\) is a fraction where both the numerator and denominator are integers and the denominator is not zero, it is a rational number.

### Conclusion for (iii):
\(\left(\frac{\sqrt{7}}{5\sqrt{2}}\right)^2\) is **rational**.

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### Final Summary:
- (i) \((2 + \sqrt{2})^2\) is **irrational**.
- (ii) \((5 + \sqrt{5})(5 - \sqrt{5})\) is **rational**.
- (iii) \(\left(\frac{\sqrt{7}}{5\sqrt{2}}\right)^2\) is **rational**.

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