Simplify `(20)/(sqrt3-sqrt(-2))+(30)/(3 sqrt(-2)-2sqrt3)-(14)/(2 sqrt3- sqrt(-2))`

Answer

To simplify the expression

\[ \frac{20}{\sqrt{3} - \sqrt{-2}} + \frac{30}{3\sqrt{-2} - 2\sqrt{3}} - \frac{14}{2\sqrt{3} - \sqrt{-2}}, \]

we will follow these steps:

### Step 1: Rewrite the square roots of negative numbers
We know that \(\sqrt{-2} = \sqrt{2}i\). Thus, we can rewrite the expression as:

\[ \frac{20}{\sqrt{3} - \sqrt{2}i} + \frac{30}{3\sqrt{2}i - 2\sqrt{3}} - \frac{14}{2\sqrt{3} - \sqrt{2}i}. \]

### Step 2: Rationalize the denominators
To simplify each term, we will rationalize the denominators by multiplying the numerator and denominator by the conjugate of the denominator.

**First term:**
\[ \frac{20}{\sqrt{3} - \sqrt{2}i} \cdot \frac{\sqrt{3} + \sqrt{2}i}{\sqrt{3} + \sqrt{2}i} = \frac{20(\sqrt{3} + \sqrt{2}i)}{(\sqrt{3})^2 - (\sqrt{2}i)^2} = \frac{20(\sqrt{3} + \sqrt{2}i)}{3 + 2} = \frac{20(\sqrt{3} + \sqrt{2}i)}{5} = 4(\sqrt{3} + \sqrt{2}i). \]

**Second term:**
\[ \frac{30}{3\sqrt{2}i - 2\sqrt{3}} \cdot \frac{3\sqrt{2}i + 2\sqrt{3}}{3\sqrt{2}i + 2\sqrt{3}} = \frac{30(3\sqrt{2}i + 2\sqrt{3})}{(3\sqrt{2}i)^2 - (2\sqrt{3})^2} = \frac{30(3\sqrt{2}i + 2\sqrt{3})}{-18 - 12} = \frac{30(3\sqrt{2}i + 2\sqrt{3})}{-30} = - (3\sqrt{2}i + 2\sqrt{3}). \]

**Third term:**
\[ -\frac{14}{2\sqrt{3} - \sqrt{2}i} \cdot \frac{2\sqrt{3} + \sqrt{2}i}{2\sqrt{3} + \sqrt{2}i} = -\frac{14(2\sqrt{3} + \sqrt{2}i)}{(2\sqrt{3})^2 - (\sqrt{2}i)^2} = -\frac{14(2\sqrt{3} + \sqrt{2}i)}{12 + 2} = -\frac{14(2\sqrt{3} + \sqrt{2}i)}{14} = -(2\sqrt{3} + \sqrt{2}i). \]

### Step 3: Combine all terms
Now we can combine all the simplified terms:

\[ 4(\sqrt{3} + \sqrt{2}i) - (3\sqrt{2}i + 2\sqrt{3}) - (2\sqrt{3} + \sqrt{2}i). \]

Distributing the negative signs:

\[ 4\sqrt{3} + 4\sqrt{2}i - 3\sqrt{2}i - 2\sqrt{3} - 2\sqrt{3} - \sqrt{2}i. \]

Combining like terms:

1. For the real parts:
\[ 4\sqrt{3} - 2\sqrt{3} - 2\sqrt{3} = 0. \]

2. For the imaginary parts:
\[ 4\sqrt{2}i - 3\sqrt{2}i - \sqrt{2}i = 0. \]

### Final Answer
Thus, the simplified expression is:

\[ 0. \]