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Simplify. Rationalize the denominator.\newline3−4+3\frac{3}{-4 + \sqrt{3}}

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Full solution

Q. Simplify. Rationalize the denominator.\newline3−4+3\frac{3}{-4 + \sqrt{3}}
  1. Select Conjugate: Select the conjugate of −4+3-4 + \sqrt{3}.\newlineConjugate of a−ba - \sqrt{b}: a+ba + \sqrt{b}\newlineConjugate of −4+3-4 + \sqrt{3}: −4−3-4 - \sqrt{3}
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it. \newline3−4+3×−4−3−4−3\frac{3}{-4 + \sqrt{3}} \times \frac{-4 - \sqrt{3}}{-4 - \sqrt{3}}
  3. Simplify Numerator: Simplify the numerator by distributing 33 to both terms in the conjugate.3×(−4)−3×33 \times (-4) - 3 \times \sqrt{3} = −12−33-12 - 3\sqrt{3}
  4. Simplify Denominator: Simplify the denominator by using the difference of squares formula.\newline(−4+3)∗(−4−3)(-4 + \sqrt{3}) * (-4 - \sqrt{3})\newline=(−4)2−(3)2= (-4)^2 - (\sqrt{3})^2\newline=16−3= 16 - 3\newline=13= 13
  5. Combine Numerator and Denominator: Combine the simplified numerator and denominator.\newline(−12−33)/13(-12 - 3\sqrt{3})/13\newlineThis fraction is already in simplest form.

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