Evaluate : `(4+3sqrt(-20))^(1//2)+(4-3 sqrt(-20))^(1//2)`

Explanation

To evaluate the expression \( (4 + 3\sqrt{-20})^{1/2} + (4 - 3\sqrt{-20})^{1/2} \), we can follow these steps:

### Step 1: Simplify the Square Roots
First, we simplify \( \sqrt{-20} \):
\[ \sqrt{-20} = \sqrt{20} \cdot \sqrt{-1} = \sqrt{20} \cdot i = 2\sqrt{5} \cdot i \]
Thus, we can rewrite the expression as:
\[ (4 + 3(2\sqrt{5}i))^{1/2} + (4 - 3(2\sqrt{5}i))^{1/2} \]
This simplifies to:
\[ (4 + 6\sqrt{5}i)^{1/2} + (4 - 6\sqrt{5}i)^{1/2} \]

### Step 2: Set Up for Comparison
Let \( z_1 = (4 + 6\sqrt{5}i)^{1/2} \) and \( z_2 = (4 - 6\sqrt{5}i)^{1/2} \). We want to find \( z_1 + z_2 \).

### Step 3: Express in Terms of Real and Imaginary Parts
Assume \( z_1 = x + yi \) and \( z_2 = u + vi \). Then:
\[ z_1^2 = 4 + 6\sqrt{5}i \quad \text{and} \quad z_2^2 = 4 - 6\sqrt{5}i \]
From \( z_1^2 = x^2 - y^2 + 2xyi \), we have:
1. \( x^2 - y^2 = 4 \)
2. \( 2xy = 6\sqrt{5} \)

From \( z_2^2 = u^2 - v^2 - 2uv i \), we have:
1. \( u^2 - v^2 = 4 \)
2. \( -2uv = -6\sqrt{5} \) or \( 2uv = 6\sqrt{5} \)

### Step 4: Solve the System of Equations
From the equations, we can combine:
1. \( x^2 - y^2 = 4 \)
2. \( u^2 - v^2 = 4 \)
3. \( 2xy = 6\sqrt{5} \)
4. \( 2uv = 6\sqrt{5} \)

### Step 5: Find \( x^2 + y^2 \) and \( u^2 + v^2 \)
Using the identity:
\[ x^2 + y^2 = \sqrt{(x^2 - y^2)^2 + (2xy)^2} \]
Substituting the values:
\[ x^2 + y^2 = \sqrt{4^2 + (6\sqrt{5})^2} = \sqrt{16 + 180} = \sqrt{196} = 14 \]

### Step 6: Solve for \( x \) and \( y \)
Now, we can add the equations:
\[ x^2 + y^2 + x^2 - y^2 = 14 + 4 \]
This simplifies to:
\[ 2x^2 = 18 \implies x^2 = 9 \implies x = \pm 3 \]
Substituting back to find \( y^2 \):
\[ 9 + y^2 = 14 \implies y^2 = 5 \implies y = \pm \sqrt{5} \]
Thus, we have:
\[ z_1 = 3 + i\sqrt{5} \quad \text{and} \quad z_2 = 3 - i\sqrt{5} \]

### Step 7: Add the Results
Adding \( z_1 \) and \( z_2 \):
\[ z_1 + z_2 = (3 + i\sqrt{5}) + (3 - i\sqrt{5}) = 6 \]

### Final Answer
Thus, the value of the original expression is:
\[ \boxed{6} \]