The simplest value of `(3sqrt(8)-2sqrt(12)+sqrt(20))/(3sqrt(18)-2sqrt(27)+sqrt(45))` is :

Options

A. A) `(3)/(2)`
B. B) `(2)/(3)`
C. C) `(1)/(3)`
D. D) `2`

Answer

To simplify the expression \((3\sqrt{8} - 2\sqrt{12} + \sqrt{20}) / (3\sqrt{18} - 2\sqrt{27} + \sqrt{45})\), we will follow these steps:

### Step 1: Simplify the square roots in the numerator
1. **Calculate \(\sqrt{8}\)**:
\[ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \]
Therefore, \(3\sqrt{8} = 3 \times 2\sqrt{2} = 6\sqrt{2}\).

2. **Calculate \(\sqrt{12}\)**:
\[ \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \]
Therefore, \(-2\sqrt{12} = -2 \times 2\sqrt{3} = -4\sqrt{3}\).

3. **Calculate \(\sqrt{20}\)**:
\[ \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \]
Therefore, \(\sqrt{20} = 2\sqrt{5}\).

Now, substituting these back into the numerator:
\[ 3\sqrt{8} - 2\sqrt{12} + \sqrt{20} = 6\sqrt{2} - 4\sqrt{3} + 2\sqrt{5} \]

### Step 2: Simplify the square roots in the denominator
1. **Calculate \(\sqrt{18}\)**:
\[ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \]
Therefore, \(3\sqrt{18} = 3 \times 3\sqrt{2} = 9\sqrt{2}\).

2. **Calculate \(\sqrt{27}\)**:
\[ \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} \]
Therefore, \(-2\sqrt{27} = -2 \times 3\sqrt{3} = -6\sqrt{3}\).

3. **Calculate \(\sqrt{45}\)**:
\[ \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} \]
Therefore, \(\sqrt{45} = 3\sqrt{5}\).

Now, substituting these back into the denominator:
\[ 3\sqrt{18} - 2\sqrt{27} + \sqrt{45} = 9\sqrt{2} - 6\sqrt{3} + 3\sqrt{5} \]

### Step 3: Combine the results
The expression now looks like this:
\[ \frac{6\sqrt{2} - 4\sqrt{3} + 2\sqrt{5}}{9\sqrt{2} - 6\sqrt{3} + 3\sqrt{5}} \]

### Step 4: Factor out common terms
We can factor out common terms from both the numerator and the denominator:
- In the numerator, we can factor out \(2\):
\[ 2(3\sqrt{2} - 2\sqrt{3} + \sqrt{5}) \]

- In the denominator, we can factor out \(3\):
\[ 3(3\sqrt{2} - 2\sqrt{3} + \sqrt{5}) \]

### Step 5: Simplify the fraction
Now we can simplify the expression:
\[ \frac{2(3\sqrt{2} - 2\sqrt{3} + \sqrt{5})}{3(3\sqrt{2} - 2\sqrt{3} + \sqrt{5})} = \frac{2}{3} \]

### Final Answer
The simplest value of the expression is:
\[ \frac{2}{3} \]