`[2 sqrt(54) - 6 sqrt((2)/(3)) - sqrt(96)]` is equal to

Options

A. 0
B. 1
C. 2
D. `sqrt(6)`

Answer

To simplify the expression \(2 \sqrt{54} - 6 \sqrt{\frac{2}{3}} - \sqrt{96}\), we will follow these steps:

### Step 1: Simplify \(\sqrt{54}\)
We can express \(54\) as \(2 \times 27\) or \(2 \times 9 \times 6\).
\[ \sqrt{54} = \sqrt{2 \times 27} = \sqrt{2 \times 9 \times 6} = \sqrt{2} \times 3 \times \sqrt{6} = 3\sqrt{2}\sqrt{6} = 3\sqrt{12} \]
Thus,
\[ 2 \sqrt{54} = 2 \times 3 \sqrt{6} = 6\sqrt{6} \]

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

### Step 2: Simplify \(\sqrt{96}\)
We can express \(96\) as \(16 \times 6\).
\[ \sqrt{96} = \sqrt{16 \times 6} = \sqrt{16} \times \sqrt{6} = 4\sqrt{6} \]

**Hint:** Look for perfect squares when simplifying square roots.

### Step 3: Simplify \(6 \sqrt{\frac{2}{3}}\)
We can rewrite this as:
\[ 6 \sqrt{\frac{2}{3}} = 6 \times \frac{\sqrt{2}}{\sqrt{3}} = \frac{6\sqrt{2}}{\sqrt{3}} \]

**Hint:** Simplify fractions under the square root separately.

### Step 4: Substitute back into the expression
Now we substitute back into the original expression:
\[ 2 \sqrt{54} - 6 \sqrt{\frac{2}{3}} - \sqrt{96} = 6\sqrt{6} - \frac{6\sqrt{2}}{\sqrt{3}} - 4\sqrt{6} \]

### Step 5: Combine like terms
Now we can combine the terms:
\[ 6\sqrt{6} - 4\sqrt{6} = 2\sqrt{6} \]
So we have:
\[ 2\sqrt{6} - \frac{6\sqrt{2}}{\sqrt{3}} \]

### Step 6: Simplify \(\frac{6\sqrt{2}}{\sqrt{3}}\)
To simplify \(\frac{6\sqrt{2}}{\sqrt{3}}\), we can multiply the numerator and denominator by \(\sqrt{3}\):
\[ \frac{6\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{6}}{3} = 2\sqrt{6} \]

### Step 7: Final simplification
Now we can substitute this back:
\[ 2\sqrt{6} - 2\sqrt{6} = 0 \]

Thus, the simplified value of the expression \(2 \sqrt{54} - 6 \sqrt{\frac{2}{3}} - \sqrt{96}\) is \(0\).

**Final Answer:** \(0\)