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Simplify. Rationalize the denominator. \newline2−2+3\frac{2}{-2 + \sqrt{3}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline2−2+3\frac{2}{-2 + \sqrt{3}}
  1. Multiply by conjugate: Multiply the numerator and the denominator by the conjugate of the denominator.\newline(2⋅(−2−3))/((−2+3)⋅(−2−3))(2 \cdot (-2 - \sqrt{3})) / ((-2 + \sqrt{3}) \cdot (-2 - \sqrt{3}))
  2. Simplify numerator: Simplify the numerator by distributing the multiplication.\newline2×(−2)+2×(−3)=−4−232 \times (-2) + 2 \times (-\sqrt{3}) = -4 - 2\sqrt{3}
  3. Use difference of squares: Simplify the denominator by using the difference of squares formula, which is (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2.(−2)2−(3)2=4−3(-2)^2 - (\sqrt{3})^2 = 4 - 3
  4. Calculate denominator: Calculate the simplified denominator. 4−3=14 - 3 = 1
  5. Final simplification: Since the denominator is now 11, the expression simplifies to the numerator.−4−23-4 - 2\sqrt{3}

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