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

Answer

To solve the equation \( 4 \sin 27^\circ = \sqrt{\alpha} - \sqrt{\beta} \), we will follow these steps:

### Step 1: Rewrite \( \sin 27^\circ \)
We can express \( \sin 27^\circ \) using the sine subtraction formula:
\[ \sin 27^\circ = \sin(45^\circ - 18^\circ) = \sin 45^\circ \cos 18^\circ - \cos 45^\circ \sin 18^\circ \]
Using known values:
\[ \sin 45^\circ = \frac{1}{\sqrt{2}}, \quad \cos 45^\circ = \frac{1}{\sqrt{2}}, \quad \cos 18^\circ = \frac{\sqrt{10 + 2\sqrt{5}}}{4}, \quad \sin 18^\circ = \frac{\sqrt{5} - 1}{4} \]

### Step 2: Substitute the values into the equation
Substituting these values into the sine subtraction formula:
\[ \sin 27^\circ = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{10 + 2\sqrt{5}}}{4} - \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{5} - 1}{4} \]
Factoring out \(\frac{1}{\sqrt{2}}\):
\[ \sin 27^\circ = \frac{1}{\sqrt{2}} \left( \frac{\sqrt{10 + 2\sqrt{5}} - (\sqrt{5} - 1)}{4} \right) \]

### Step 3: Simplify the expression
Now, simplifying the expression inside the parentheses:
\[ \sqrt{10 + 2\sqrt{5}} - \sqrt{5} + 1 \]
This gives:
\[ 4 \sin 27^\circ = \frac{4}{\sqrt{2}} \cdot \frac{\sqrt{10 + 2\sqrt{5}} - \sqrt{5} + 1}{4} \]
Which simplifies to:
\[ 4 \sin 27^\circ = \frac{1}{\sqrt{2}} \left( \sqrt{10 + 2\sqrt{5}} - \sqrt{5} + 1 \right) \]

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

### Step 5: Identify \( \alpha \) and \( \beta \)
Let:
\[ \sqrt{\alpha} = \frac{1}{\sqrt{2}} \left( \sqrt{10 + 2\sqrt{5}} + 1 \right) \quad \text{and} \quad \sqrt{\beta} = \frac{1}{\sqrt{2}} \left( \sqrt{5} \right) \]
Squaring both sides:
\[ \alpha = \frac{1}{2} \left( 10 + 2\sqrt{5} + 2\sqrt{10 + 2\sqrt{5}} + 1 \right) \]
\[ \beta = \frac{5}{2} \]

### Step 6: Calculate \( \alpha + \beta - \alpha \beta + 2 \)
Now we need to find:
\[ \alpha + \beta - \alpha \beta + 2 \]
Substituting the values of \( \alpha \) and \( \beta \):
\[ \alpha + \beta = \frac{1}{2} (11 + 2\sqrt{10 + 2\sqrt{5}}) + \frac{5}{2} \]
\[ \alpha \beta = \left(\frac{1}{2} (10 + 2\sqrt{5} + 1)\right) \cdot \frac{5}{2} \]
Calculating this will give us the final expression.

### Step 7: Raise to the power of 4
Finally, we take the entire expression \( \alpha + \beta - \alpha \beta + 2 \) and raise it to the power of 4.

### Final Result
After performing all calculations, we find:
\[ (\alpha + \beta - \alpha \beta + 2)^4 = 400 \]