It looks like you want to simplify the expression \( \frac{2}{5 - \sqrt{3}} \) and rationalize the denominator. Here are the steps to achieve that:
- Multiply by the conjugate of the denominator: The conjugate of \( 5 - \sqrt{3} \) is \( 5 + \sqrt{3} \). We will multiply the numerator and the denominator by this conjugate.
\[ \frac{2}{5 - \sqrt{3}} \cdot \frac{5 + \sqrt{3}}{5 + \sqrt{3}} \]
- Perform the multiplication:
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In the numerator: \[ 2(5 + \sqrt{3}) = 10 + 2\sqrt{3} \]
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In the denominator: \[ (5 - \sqrt{3})(5 + \sqrt{3}) = 5^2 - (\sqrt{3})^2 = 25 - 3 = 22 \]
- Combine the results:
\[ \frac{10 + 2\sqrt{3}}{22} \]
- Simplify the expression:
We can simplify this fraction by dividing both terms in the numerator by 2:
\[ = \frac{5 + \sqrt{3}}{11} \]
So, the simplified and rationalized form of the original expression \( \frac{2}{5 - \sqrt{3}} \) is
\[ \frac{5 + \sqrt{3}}{11} \]
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