Simplify : `sqrt(192) - 1/2 sqrt(48) - sqrt(75)`.

Answer

To simplify the expression \( \sqrt{192} - \frac{1}{2} \sqrt{48} - \sqrt{75} \), we will follow these steps:

### Step 1: Simplify \( \sqrt{192} \)
First, we will factor 192 into its prime factors:
\[ 192 = 2^6 \times 3 \]
Now, we can simplify \( \sqrt{192} \):
\[ \sqrt{192} = \sqrt{2^6 \times 3} = \sqrt{(2^3)^2 \times 3} = 2^3 \sqrt{3} = 8\sqrt{3} \]

### Step 2: Simplify \( \sqrt{48} \)
Next, we will factor 48:
\[ 48 = 2^4 \times 3 \]
Now, we can simplify \( \sqrt{48} \):
\[ \sqrt{48} = \sqrt{2^4 \times 3} = \sqrt{(2^2)^2 \times 3} = 2^2 \sqrt{3} = 4\sqrt{3} \]

### Step 3: Simplify \( \sqrt{75} \)
Now, we will factor 75:
\[ 75 = 3 \times 5^2 \]
Now, we can simplify \( \sqrt{75} \):
\[ \sqrt{75} = \sqrt{3 \times 5^2} = 5\sqrt{3} \]

### Step 4: Substitute back into the expression
Now we substitute the simplified forms back into the original expression:
\[ \sqrt{192} - \frac{1}{2} \sqrt{48} - \sqrt{75} = 8\sqrt{3} - \frac{1}{2}(4\sqrt{3}) - 5\sqrt{3} \]

### Step 5: Simplify the expression
Calculate \( \frac{1}{2}(4\sqrt{3}) \):
\[ \frac{1}{2}(4\sqrt{3}) = 2\sqrt{3} \]
Now substitute this back:
\[ 8\sqrt{3} - 2\sqrt{3} - 5\sqrt{3} \]
Combine like terms:
\[ (8 - 2 - 5)\sqrt{3} = 1\sqrt{3} = \sqrt{3} \]

### Final Answer
Thus, the simplified expression is:
\[ \sqrt{3} \]