Examine whether the following numbers are rational or irrational :
` (3- sqrt5) ^(2)`

Answer

To determine whether the number \((3 - \sqrt{5})^2\) is rational or irrational, we can follow these steps:

### Step 1: Expand the expression
We start by expanding the expression \((3 - \sqrt{5})^2\) using the formula \((a - b)^2 = a^2 - 2ab + b^2\).

\[ (3 - \sqrt{5})^2 = 3^2 - 2 \cdot 3 \cdot \sqrt{5} + (\sqrt{5})^2 \]

### Step 2: Calculate each term
Now we calculate each term in the expansion:

- \(3^2 = 9\)
- \((\sqrt{5})^2 = 5\)
- \(-2 \cdot 3 \cdot \sqrt{5} = -6\sqrt{5}\)

Putting it all together, we have:

\[ (3 - \sqrt{5})^2 = 9 - 6\sqrt{5} + 5 \]

### Step 3: Combine like terms
Now we combine the constant terms:

\[ 9 + 5 = 14 \]

So, we can rewrite the expression as:

\[ (3 - \sqrt{5})^2 = 14 - 6\sqrt{5} \]

### Step 4: Analyze the expression
Now we need to determine if \(14 - 6\sqrt{5}\) is rational or irrational.

- The number \(14\) is rational.
- The term \(-6\sqrt{5}\) involves \(\sqrt{5}\), which is known to be an irrational number.

### Step 5: Conclusion
Since we are subtracting an irrational number (\(-6\sqrt{5}\)) from a rational number (14), the result \(14 - 6\sqrt{5}\) will also be irrational.

Therefore, we conclude that:

\[ (3 - \sqrt{5})^2 \text{ is an irrational number.} \]