Question

1. rationalize the denominator and simplify.

\\(\frac{5 + \sqrt{3}}{2 - \sqrt{3}}\\)
options:
\\(\frac{5 + \sqrt{3}}{2}\\)
\\(-\frac{2}{5}\\)
\\(-13 - 7\sqrt{3}\\)
\\(13 + 7\sqrt{3}\\)

Explanation:

Step1: Multiply numerator and denominator by the conjugate of the denominator.

The conjugate of \(2 - \sqrt{3}\) is \(2 + \sqrt{3}\). So we multiply the fraction \(\frac{5 + \sqrt{3}}{2 - \sqrt{3}}\) by \(\frac{2 + \sqrt{3}}{2 + \sqrt{3}}\):

$$ \frac{(5 + \sqrt{3})(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})} $$

Step2: Expand the numerator and the denominator.

• Expand the numerator using the distributive property (FOIL method):

$$
LATEXBLOCK0
$$

• Expand the denominator using the difference of squares formula \((a - b)(a + b)=a^{2}-b^{2}\), where \(a = 2\) and \(b=\sqrt{3}\):

$$ (2 - \sqrt{3})(2 + \sqrt{3})=2^{2}-(\sqrt{3})^{2}=4 - 3 = 1 $$

Step3: Simplify the fraction.

Since the denominator is \(1\), the fraction simplifies to the numerator: \(13 + 7\sqrt{3}\)

Answer:

\(13 + 7\sqrt{3}\)