Question

question 4

simplify the expression.

\\\frac{-3+\sqrt{3}}{2-\sqrt{7}}\\

• \\(\frac{-7+7\sqrt{6}-7\sqrt{2}+14\sqrt{3}}{5}\\)
• \\(\frac{6+3\sqrt{7}-2\sqrt{3}-\sqrt{21}}{3}\\)
• \\(\frac{8+2\sqrt{5}-24\sqrt{3}-6\sqrt{15}}{11}\\)
• \\(16-10\sqrt{3}\\)

Explanation:

⚡ Using what you learned: Rationalizing Denominators

Step 1: Identify the conjugate

Multiply the numerator and the denominator by the conjugate of the denominator, \(2 + \sqrt{7}\):

$$ \frac{-3 + \sqrt{3}}{2 - \sqrt{7}} \cdot \frac{2 + \sqrt{7}}{2 + \sqrt{7}} $$

Step 2: Expand the numerator

$$ (-3 + \sqrt{3})(2 + \sqrt{7}) = -6 - 3\sqrt{7} + 2\sqrt{3} + \sqrt{21} $$

Step 3: Expand the denominator

$$ (2 - \sqrt{7})(2 + \sqrt{7}) = 2^2 - (\sqrt{7})^2 = 4 - 7 = -3 $$

Step 4: Simplify the fraction

$$ \frac{-6 - 3\sqrt{7} + 2\sqrt{3} + \sqrt{21}}{-3} = \frac{6 + 3\sqrt{7} - 2\sqrt{3} - \sqrt{21}}{3} $$

Answer:

$$ \frac{6+3\sqrt{7}-2\sqrt{3}-\sqrt{21}}{3} $$