State, whether the following numbers are rational or not:
`(5+sqrt(5)) ( 5- sqrt(5))`

Answer

To determine whether the number \((5 + \sqrt{5})(5 - \sqrt{5})\) is rational or not, we can follow these steps:

### Step 1: Recognize the expression
We have the expression \((5 + \sqrt{5})(5 - \sqrt{5})\). This is a product of two binomials.

**Hint:** Look for a way to simplify the expression using algebraic identities.

### Step 2: Apply the difference of squares formula
We can use the difference of squares formula, which states that \((a + b)(a - b) = a^2 - b^2\). Here, let \(a = 5\) and \(b = \sqrt{5}\).

**Hint:** Identify \(a\) and \(b\) in the expression to apply the formula correctly.

### Step 3: Calculate \(a^2\) and \(b^2\)
Now, we calculate:
- \(a^2 = 5^2 = 25\)
- \(b^2 = (\sqrt{5})^2 = 5\)

**Hint:** Remember that squaring a square root gives you the number inside the square root.

### Step 4: Substitute back into the formula
Using the difference of squares formula:
\[ (5 + \sqrt{5})(5 - \sqrt{5}) = a^2 - b^2 = 25 - 5 \]

**Hint:** Make sure to perform the subtraction correctly.

### Step 5: Simplify the result
Now, simplify \(25 - 5\):
\[ 25 - 5 = 20 \]

**Hint:** Check if the result is a whole number, as whole numbers are rational.

### Step 6: Conclusion
Since \(20\) is a whole number, it is also a rational number. Therefore, we conclude that \((5 + \sqrt{5})(5 - \sqrt{5})\) is a rational number.

**Final Answer:** Yes, \((5 + \sqrt{5})(5 - \sqrt{5})\) is a rational number.

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