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Simplify. Rationalize the denominator. \newline5−6+5\frac{5}{-6 + \sqrt{5}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline5−6+5\frac{5}{-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 both the numerator and the denominator by the conjugate of the denominator.\newline5−6+5×−6−5−6−5\frac{5}{-6 + \sqrt{5}} \times \frac{-6 - \sqrt{5}}{-6 - \sqrt{5}}
  3. Simplify Numerator: Simplify the numerator.\newlineMultiply 55 by the conjugate −6−5-6 - \sqrt{5}.\newline5×(−6)−5×55 \times (-6) - 5 \times \sqrt{5}\newline=−30−55= -30 - 5\sqrt{5}
  4. Simplify Denominator: Simplify the denominator.\newlineSimplify (−6+5)∗(−6−5)(-6 + \sqrt{5}) * (-6 - \sqrt{5}).\newline(−6)2−(5)2(-6)^2 - (\sqrt{5})^2\newline= 36−536 - 5\newline= 3131
  5. Combine Numerator and Denominator: Combine the simplified numerator and denominator. \newline(−30−55)/31(-30 - 5\sqrt{5})/31\newlineThis fraction is already in simplest form.

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