Find the fraction equivalent to `5/(2-sqrt3)`, such that denominator of the fraction is not irrational.

Answer

To find the fraction equivalent to \( \frac{5}{2 - \sqrt{3}} \) such that the denominator is not irrational, we will use the method of rationalization. Here’s the step-by-step solution:

### Step 1: Identify the expression
We start with the expression:
\[ \frac{5}{2 - \sqrt{3}} \]

### Step 2: Multiply by the conjugate
To eliminate the irrational part in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \( 2 + \sqrt{3} \):
\[ \frac{5}{2 - \sqrt{3}} \cdot \frac{2 + \sqrt{3}}{2 + \sqrt{3}} \]

### Step 3: Simplify the numerator
Now, we multiply the numerators:
\[ 5 \cdot (2 + \sqrt{3}) = 10 + 5\sqrt{3} \]

### Step 4: Simplify the denominator
Next, we simplify the denominator using the difference of squares formula:
\[ (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

### Step 5: Combine the results
Now we can combine the results:
\[ \frac{10 + 5\sqrt{3}}{1} = 10 + 5\sqrt{3} \]

### Final Result
Thus, the fraction equivalent to \( \frac{5}{2 - \sqrt{3}} \) with a rational denominator is:
\[ 10 + 5\sqrt{3} \]