Perform the indicated operation and give your answer in the form `x+yi`, where x and y are real numbers and `i= sqrt(-1)`
`(5+2i)/(-1+sqrt3i)`

Answer

To solve the problem \((5 + 2i) / (-1 + \sqrt{3}i)\) and express the answer in the form \(x + yi\), we follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ z = \frac{5 + 2i}{-1 + \sqrt{3}i} \]

### Step 2: Multiply by the conjugate of the denominator
To eliminate the imaginary part in the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \(-1 - \sqrt{3}i\):
\[ z = \frac{(5 + 2i)(-1 - \sqrt{3}i)}{(-1 + \sqrt{3}i)(-1 - \sqrt{3}i)} \]

### Step 3: Expand the numerator
Now we expand the numerator:
\[ (5 + 2i)(-1 - \sqrt{3}i) = 5(-1) + 5(-\sqrt{3}i) + 2i(-1) + 2i(-\sqrt{3}i) \]
Calculating each term:
- \(5(-1) = -5\)
- \(5(-\sqrt{3}i) = -5\sqrt{3}i\)
- \(2i(-1) = -2i\)
- \(2i(-\sqrt{3}i) = -2\sqrt{3}i^2 = 2\sqrt{3}\) (since \(i^2 = -1\))

Combining these, we get:
\[ -5 + 2\sqrt{3} - (5\sqrt{3} + 2)i = (2\sqrt{3} - 5) + (-5\sqrt{3} - 2)i \]

### Step 4: Expand the denominator
Now we expand the denominator using the difference of squares:
\[ (-1 + \sqrt{3}i)(-1 - \sqrt{3}i) = (-1)^2 - (\sqrt{3}i)^2 = 1 - 3(-1) = 1 + 3 = 4 \]

### Step 5: Combine the results
Now we can write the expression for \(z\):
\[ z = \frac{(2\sqrt{3} - 5) + (-5\sqrt{3} - 2)i}{4} \]

### Step 6: Separate into real and imaginary parts
We can separate the real and imaginary parts:
\[ z = \frac{2\sqrt{3} - 5}{4} + \frac{-5\sqrt{3} - 2}{4}i \]

### Step 7: Final answer
Thus, the final answer in the form \(x + yi\) is:
\[ z = \frac{2\sqrt{3} - 5}{4} + \frac{-5\sqrt{3} - 2}{4}i \]