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Simplify. Rationalize the denominator.\newline4−8−5\frac{4}{-8 - \sqrt{5}}

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Full solution

Q. Simplify. Rationalize the denominator.\newline4−8−5\frac{4}{-8 - \sqrt{5}}
  1. Select Conjugate: Select the conjugate of −8−5-8 - \sqrt{5}.\newlineConjugate of a−ba - \sqrt{b}: a+ba + \sqrt{b}\newlineConjugate of −8−5-8 - \sqrt{5}: −8+5-8 + \sqrt{5}
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it.\newline4(−8−5)×(−8+5)(−8+5)\frac{4}{(-8 - \sqrt{5})} \times \frac{(-8 + \sqrt{5})}{(-8 + \sqrt{5})}
  3. Simplify Numerator: Simplify the numerator by distributing the 44 across the conjugate −8+5-8 + \sqrt{5}.4×(−8)+4×54 \times (-8) + 4 \times \sqrt{5}=−32+45= -32 + 4\sqrt{5}
  4. Simplify Denominator: Simplify the denominator by using the difference of squares formula: (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2.\newline(−8)2−(5)2(-8)^2 - (\sqrt{5})^2\newline= 6464 - 55\newline= 5959
  5. Combine Numerator and Denominator: Combine the simplified numerator and denominator. \newline(−32+45)/59(-32 + 4\sqrt{5})/59\newlineThis fraction is already in simplest form.

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