The simplest value of `(3sqrt8 - 2 sqrt12 + sqrt20)/(3 sqrt18 - 2 sqrt 27 + sqrt45)`

Options

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

Answer

To solve the expression \((3\sqrt{8} - 2\sqrt{12} + \sqrt{20})/(3\sqrt{18} - 2\sqrt{27} + \sqrt{45})\), we will simplify both the numerator and the denominator step by step.

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

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

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

Putting these together, the numerator becomes:
\[ 6\sqrt{2} - 4\sqrt{3} + 2\sqrt{5} \]

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

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

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

Putting these together, the denominator becomes:
\[ 9\sqrt{2} - 6\sqrt{3} + 3\sqrt{5} \]

### Step 3: Write the simplified expression
Now we can write the simplified expression:
\[ \frac{6\sqrt{2} - 4\sqrt{3} + 2\sqrt{5}}{9\sqrt{2} - 6\sqrt{3} + 3\sqrt{5}} \]

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

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

### Step 5: Simplify the fraction
Now we can simplify the fraction:
\[ \frac{2(3\sqrt{2} - 2\sqrt{3} + \sqrt{5})}{3(3\sqrt{2} - 2\sqrt{3} + \sqrt{5})} \]
The common term \(3\sqrt{2} - 2\sqrt{3} + \sqrt{5}\) cancels out, giving us:
\[ \frac{2}{3} \]

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