Simplify : `sqrt(18)/(5sqrt(18)+3sqrt(72)-2sqrt(162))`

Answer

To simplify the expression \(\frac{\sqrt{18}}{5\sqrt{18} + 3\sqrt{72} - 2\sqrt{162}}\), we will follow these steps:

### Step 1: Simplify the square roots in the expression
- \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\)
- \(\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\)
- \(\sqrt{162} = \sqrt{81 \times 2} = 9\sqrt{2}\)

### Step 2: Substitute the simplified square roots back into the expression
Now, we can substitute these values back into the expression:
\[ \frac{\sqrt{18}}{5\sqrt{18} + 3\sqrt{72} - 2\sqrt{162}} = \frac{3\sqrt{2}}{5(3\sqrt{2}) + 3(6\sqrt{2}) - 2(9\sqrt{2})} \]

### Step 3: Simplify the denominator
Calculating the denominator:
- \(5(3\sqrt{2}) = 15\sqrt{2}\)
- \(3(6\sqrt{2}) = 18\sqrt{2}\)
- \(-2(9\sqrt{2}) = -18\sqrt{2}\)

Now, combine these:
\[ 15\sqrt{2} + 18\sqrt{2} - 18\sqrt{2} = 15\sqrt{2} \]

### Step 4: Rewrite the expression
Now, we can rewrite the expression as:
\[ \frac{3\sqrt{2}}{15\sqrt{2}} \]

### Step 5: Cancel out \(\sqrt{2}\)
Since \(\sqrt{2}\) is present in both the numerator and the denominator, we can cancel it out:
\[ \frac{3}{15} \]

### Step 6: Simplify the fraction
Now simplify \(\frac{3}{15}\):
\[ \frac{3}{15} = \frac{1}{5} \]

### Final Answer
Thus, the simplified form of the expression is:
\[ \frac{1}{5} \]