Rationalize the denominator :
`14/(5sqrt(3)-sqrt(5))`

Answer

To rationalize the denominator of the expression \( \frac{14}{5\sqrt{3} - \sqrt{5}} \), we will follow these steps:

### Step 1: Identify the conjugate of the denominator
The denominator is \( 5\sqrt{3} - \sqrt{5} \). The conjugate of this expression is \( 5\sqrt{3} + \sqrt{5} \).

### Step 2: Multiply the numerator and denominator by the conjugate
We multiply both the numerator and the denominator by the conjugate:
\[ \frac{14}{5\sqrt{3} - \sqrt{5}} \cdot \frac{5\sqrt{3} + \sqrt{5}}{5\sqrt{3} + \sqrt{5}} \]

### Step 3: Simplify the numerator
The numerator becomes:
\[ 14(5\sqrt{3} + \sqrt{5}) = 70\sqrt{3} + 14\sqrt{5} \]

### Step 4: Simplify the denominator using the difference of squares
The denominator becomes:
\[ (5\sqrt{3})^2 - (\sqrt{5})^2 = 25 \cdot 3 - 5 = 75 - 5 = 70 \]

### Step 5: Write the final expression
Now we can write the expression as:
\[ \frac{70\sqrt{3} + 14\sqrt{5}}{70} \]

### Step 6: Simplify the fraction
We can simplify this further:
\[ \frac{70\sqrt{3}}{70} + \frac{14\sqrt{5}}{70} = \sqrt{3} + \frac{1}{5}\sqrt{5} \]

Thus, the final answer is:
\[ \sqrt{3} + \frac{1}{5}\sqrt{5} \]

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