If `sqrt(3)` = 1.7321, the value of `sqrt(192) - (1)/(2) sqrt(48) - sqrt(75)`, correct to 3 places of decimal, is

Options

A. 8.661
B. 4.331
C. 1.7321
D. -1.732

Answer

To solve the expression \( \sqrt{192} - \frac{1}{2} \sqrt{48} - \sqrt{75} \) given that \( \sqrt{3} = 1.7321 \), we will simplify each term step by step.

### Step 1: Simplify \( \sqrt{192} \)
We can express \( 192 \) as \( 64 \times 3 \):
\[ \sqrt{192} = \sqrt{64 \times 3} = \sqrt{64} \times \sqrt{3} = 8\sqrt{3} \]

**Hint:** Factor the number under the square root into perfect squares to simplify.

### Step 2: Simplify \( \sqrt{48} \)
We can express \( 48 \) as \( 16 \times 3 \):
\[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \]

**Hint:** Look for perfect squares that can be factored out of the number under the square root.

### Step 3: Simplify \( \sqrt{75} \)
We can express \( 75 \) as \( 25 \times 3 \):
\[ \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \]

**Hint:** Again, factor into perfect squares to simplify the square root.

### Step 4: Substitute back into the expression
Now we can 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
Calculating \( \frac{1}{2}(4\sqrt{3}) \):
\[ \frac{1}{2}(4\sqrt{3}) = 2\sqrt{3} \]
Now substitute this back into the expression:
\[ 8\sqrt{3} - 2\sqrt{3} - 5\sqrt{3} \]
Combine the terms:
\[ (8 - 2 - 5)\sqrt{3} = 1\sqrt{3} = \sqrt{3} \]

### Step 6: Substitute the value of \( \sqrt{3} \)
Now we substitute \( \sqrt{3} = 1.7321 \):
\[ \sqrt{3} = 1.7321 \]

### Final Answer
Thus, the value of the expression \( \sqrt{192} - \frac{1}{2} \sqrt{48} - \sqrt{75} \) is:
\[ 1.7321 \]

### Rounding to 3 decimal places:
The final answer rounded to three decimal places is:
\[ \text{Answer} = 1.732 \]