Question

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1. \\(\frac{2}{4 - \sqrt{3}}\\)

Explanation:

Step1: Rationalize the denominator

To rationalize the denominator of \(\frac{2}{4 - \sqrt{3}}\), we multiply the numerator and denominator by the conjugate of the denominator, which is \(4 + \sqrt{3}\).

$$ \frac{2}{4 - \sqrt{3}} \times \frac{4 + \sqrt{3}}{4 + \sqrt{3}} $$

Step2: Multiply the numerators and denominators

Multiply the numerators: \(2\times(4 + \sqrt{3}) = 8 + 2\sqrt{3}\)
Multiply the denominators using the difference of squares formula \((a - b)(a + b)=a^2 - b^2\), where \(a = 4\) and \(b=\sqrt{3}\):

$$ (4 - \sqrt{3})(4 + \sqrt{3})=4^2-(\sqrt{3})^2 = 16 - 3=13 $$

So the fraction becomes \(\frac{8 + 2\sqrt{3}}{13}\)

Answer:

\(\frac{8 + 2\sqrt{3}}{13}\)