Find the value of `(6sqrt2)/(5sqrt3)(sqrt2 + sqrt18)`

Options

A. `(16sqrt3)/(5)`
B. `(26sqrt3)/(5)`
C. `(36sqrt3)/(5)`
D. `(46sqrt3)/(5)`

Answer

To solve the expression \(\frac{6\sqrt{2}}{5\sqrt{3}(\sqrt{2} + \sqrt{18})}\), we can follow these steps:

### Step 1: Simplify \(\sqrt{18}\)
First, we simplify \(\sqrt{18}\):
\[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \]

### Step 2: Substitute \(\sqrt{18}\) back into the expression
Now, we substitute \(\sqrt{18}\) back into the expression:
\[ \sqrt{2} + \sqrt{18} = \sqrt{2} + 3\sqrt{2} = 4\sqrt{2} \]

### Step 3: Rewrite the original expression
Now, we can rewrite the original expression:
\[ \frac{6\sqrt{2}}{5\sqrt{3}(4\sqrt{2})} \]

### Step 4: Simplify the expression
Next, we simplify the expression:
\[ \frac{6\sqrt{2}}{5\sqrt{3} \cdot 4\sqrt{2}} = \frac{6\sqrt{2}}{20\sqrt{3}\sqrt{2}} \]
Since \(\sqrt{2}\) is in both the numerator and the denominator, we can cancel it out:
\[ = \frac{6}{20\sqrt{3}} = \frac{3}{10\sqrt{3}} \]

### Step 5: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and the denominator by \(\sqrt{3}\):
\[ = \frac{3\sqrt{3}}{10 \cdot 3} = \frac{3\sqrt{3}}{30} = \frac{\sqrt{3}}{10} \]

### Final Answer
Thus, the final value of the expression is:
\[ \frac{\sqrt{3}}{10} \]

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