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Simplify. Rationalize the denominator.\newline10−10+5\frac{10}{-10 + \sqrt{5}}

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Full solution

Q. Simplify. Rationalize the denominator.\newline10−10+5\frac{10}{-10 + \sqrt{5}}
  1. Select Conjugate: Select the conjugate of −10+5-10 + \sqrt{5}.\newlineConjugate of a−ba - \sqrt{b}: a+ba + \sqrt{b}\newlineConjugate of −10+5-10 + \sqrt{5}: −10−5-10 - \sqrt{5}
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator.\newlineWhich expression can be used to rationalize the denominator?\newlineMultiply (−10−5)(-10 - \sqrt{5}) with 1010, and −10+5-10 + \sqrt{5}.\newline10⋅(−10−5)(−10+5)⋅(−10−5)\frac{10 \cdot (-10 - \sqrt{5})}{(-10 + \sqrt{5}) \cdot (-10 - \sqrt{5})}
  3. Simplify Numerator: Simplify the numerator: 10×(−10−5)10 \times (-10 - \sqrt{5})\newline10×(−10)−10×(5)10 \times (-10) - 10 \times (\sqrt{5})\newline=−100−10×5= -100 - 10 \times \sqrt{5}
  4. Simplify Denominator: Simplify the denominator: (−10+5)∗(−10−5)(-10 + \sqrt{5}) * (-10 - \sqrt{5})\newline(−10)2−(5)2(-10)^2 - (\sqrt{5})^2\newline=100−5= 100 - 5\newline$= \(95\)
  5. Combine Numerator and Denominator: Combine the simplified numerator and denominator.\(\newline\)\((-100 - 10 \cdot \sqrt{5}) / 95\)\(\newline\)This fraction is already in simplest form.

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