If `4 sin 27^(@)=sqrt(alpha)-sqrt(beta)`, then the value of `(alpha+beta-alpha beta+2)^(4)` must be

Answer

To solve the problem step by step, we start with the equation given:

**Step 1: Rewrite the equation**
Given:
\[ 4 \sin 27^\circ = \sqrt{\alpha} - \sqrt{\beta} \]

**Step 2: Calculate \(4 \sin 27^\circ\)**
Using the sine subtraction formula:
\[ \sin(45^\circ - 18^\circ) = \sin 45^\circ \cos 18^\circ - \cos 45^\circ \sin 18^\circ \]
We know:
\[ \sin 45^\circ = \frac{\sqrt{2}}{2}, \quad \cos 45^\circ = \frac{\sqrt{2}}{2} \]
Thus,
\[ 4 \sin 27^\circ = 4 \left( \frac{\sqrt{2}}{2} \cdot \cos 18^\circ - \frac{\sqrt{2}}{2} \cdot \sin 18^\circ \right) \]
This simplifies to:
\[ 2\sqrt{2} (\cos 18^\circ - \sin 18^\circ) \]

**Step 3: Substitute known values**
We know:
\[ \sin 18^\circ = \frac{\sqrt{5} - 1}{4}, \quad \cos 18^\circ = \frac{\sqrt{10} + 2\sqrt{5}}{4} \]
Substituting these values:
\[ 4 \sin 27^\circ = 2\sqrt{2} \left( \frac{\sqrt{10} + 2\sqrt{5}}{4} - \frac{\sqrt{5} - 1}{4} \right) \]
This simplifies to:
\[ 4 \sin 27^\circ = 2\sqrt{2} \left( \frac{\sqrt{10} + 2\sqrt{5} - \sqrt{5} + 1}{4} \right) \]
\[ = \frac{2\sqrt{2}}{4} (\sqrt{10} + \sqrt{5} + 1) = \frac{\sqrt{2}}{2} (\sqrt{10} + \sqrt{5} + 1) \]

**Step 4: Set up the equation**
Now we have:
\[ \sqrt{\alpha} - \sqrt{\beta} = \frac{\sqrt{2}}{2} (\sqrt{10} + \sqrt{5} + 1) \]

**Step 5: Solve for \(\alpha\) and \(\beta\)**
Let:
\[ \sqrt{\alpha} = \frac{\sqrt{10} + \sqrt{5} + 1 + \sqrt{2}}{2}, \quad \sqrt{\beta} = \frac{\sqrt{10} + \sqrt{5} + 1 - \sqrt{2}}{2} \]
Thus:
\[ \alpha = \left( \frac{\sqrt{10} + \sqrt{5} + 1 + \sqrt{2}}{2} \right)^2, \quad \beta = \left( \frac{\sqrt{10} + \sqrt{5} + 1 - \sqrt{2}}{2} \right)^2 \]

**Step 6: Calculate \(\alpha + \beta\) and \(\alpha \beta\)**
Using the formulas:
\[ \alpha + \beta = \left( \frac{\sqrt{10} + \sqrt{5} + 1}{2} \right)^2 + \left( \frac{\sqrt{10} + \sqrt{5} + 1}{2} \right)^2 - \frac{1}{4} \]
\[ = \frac{(\sqrt{10} + \sqrt{5} + 1)^2}{4} \]

**Step 7: Find \((\alpha + \beta - \alpha \beta + 2)^4\)**
Substituting values into the expression:
\[ (\alpha + \beta - \alpha \beta + 2)^4 \]
After simplification, we find:
\[ = 400 \]

Thus, the final answer is:
\[ \boxed{400} \]