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Simplify. Rationalize the denominator. \newline9−5+5\frac{9}{-5 + \sqrt{5}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline9−5+5\frac{9}{-5 + \sqrt{5}}
  1. Find Conjugate: Select the conjugate of −5+5-5 + \sqrt{5}.\newlineConjugate of a−ba - \sqrt{b}: a+ba + \sqrt{b}\newlineConjugate of −5+5-5 + \sqrt{5}: −5−5-5 - \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 expression by (−5−5)/(−5−5)(-5 - \sqrt{5})/(-5 - \sqrt{5}).\newline9(−5+5)×(−5−5)(−5−5)\frac{9}{(-5 + \sqrt{5})} \times \frac{(-5 - \sqrt{5})}{(-5 - \sqrt{5})}
  3. Simplify Numerator: Simplify the numerator by distributing the multiplication.\newline9×(−5−5)9 \times (-5 - \sqrt{5})\newline= 9×(−5)+9×(−5)9 \times (-5) + 9 \times (-\sqrt{5})\newline= −45−95-45 - 9\sqrt{5}
  4. Simplify Denominator: Simplify the denominator by using the difference of squares formula.\newline(−5+5)∗(−5−5)(-5 + \sqrt{5}) * (-5 - \sqrt{5})\newline=(−5)2−(5)2= (-5)^2 - (\sqrt{5})^2\newline=25−5= 25 - 5\newline=20= 20
  5. Write Simplified Expression: Write the simplified expression.\newline(−45−95)/20(-45 - 9 \sqrt{5})/20\newlineThis fraction is already in simplest form.

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