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Simplify. Rationalize the denominator. \newline6−6+5\frac{6}{-6 + \sqrt{5}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline6−6+5\frac{6}{-6 + \sqrt{5}}
  1. Find Conjugate: Select the conjugate of −6+5-6 + \sqrt{5}.\newlineConjugate of a−ba - \sqrt{b}: a+ba + \sqrt{b}\newlineConjugate of −6+5-6 + \sqrt{5}: −6−5-6 - \sqrt{5}
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator.\newlineTo rationalize the denominator, we multiply the fraction by (−6−5)/(−6−5)(-6 - \sqrt{5})/(-6 - \sqrt{5}).\newline6−6+5×−6−5−6−5\frac{6}{-6 + \sqrt{5}} \times \frac{-6 - \sqrt{5}}{-6 - \sqrt{5}}
  3. Simplify Numerator: Simplify the numerator by distributing the 66.6×(−6−5)=6×(−6)+6×(−5)=−36−656 \times (-6 - \sqrt{5}) = 6 \times (-6) + 6 \times (-\sqrt{5}) = -36 - 6\sqrt{5}
  4. Simplify Denominator: Simplify the denominator using the difference of squares formula.\newline(−6+5)∗(−6−5)(-6 + \sqrt{5}) * (-6 - \sqrt{5})\newline=(−6)2−(5)2= (-6)^2 - (\sqrt{5})^2\newline=36−5= 36 - 5\newline=31= 31
  5. Combine Numerator and Denominator: Combine the simplified numerator and denominator.\newline(−36−65)/31(-36 - 6\sqrt{5})/31\newlineThis fraction is already in simplest form.

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