What is the value of `((1)/(2)+isqrt(5))((1)/(2)-isqrt(5))`?

Answer

To solve the expression \(\left(\frac{1}{2} + i\sqrt{5}\right)\left(\frac{1}{2} - i\sqrt{5}\right)\), we can use the difference of squares formula, which states that \(a^2 - b^2 = (a + b)(a - b)\).

### Step-by-Step Solution:

1. **Identify \(a\) and \(b\)**:
- Let \(a = \frac{1}{2}\)
- Let \(b = i\sqrt{5}\)

2. **Apply the difference of squares formula**:
\[ \left(\frac{1}{2} + i\sqrt{5}\right)\left(\frac{1}{2} - i\sqrt{5}\right) = a^2 - b^2 \]

3. **Calculate \(a^2\)**:
\[ a^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

4. **Calculate \(b^2\)**:
\[ b^2 = (i\sqrt{5})^2 = i^2 \cdot (\sqrt{5})^2 = -1 \cdot 5 = -5 \]

5. **Combine \(a^2\) and \(b^2\)**:
\[ a^2 - b^2 = \frac{1}{4} - (-5) = \frac{1}{4} + 5 \]

6. **Convert 5 to a fraction**:
\[ 5 = \frac{20}{4} \]

7. **Add the fractions**:
\[ \frac{1}{4} + \frac{20}{4} = \frac{1 + 20}{4} = \frac{21}{4} \]

### Final Answer:
The value of \(\left(\frac{1}{2} + i\sqrt{5}\right)\left(\frac{1}{2} - i\sqrt{5}\right)\) is \(\frac{21}{4}\).