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Simplify. Rationalize the denominator. \newline9−5+2\frac{9}{-5 + \sqrt{2}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline9−5+2\frac{9}{-5 + \sqrt{2}}
  1. Identify Conjugate of Denominator: Identify the conjugate of the denominator −5+2-5 + \sqrt{2}. The conjugate of a number of the form a+ba + \sqrt{b} is a−ba - \sqrt{b}. Therefore, the conjugate of −5+2-5 + \sqrt{2} is −5−2-5 - \sqrt{2}.
  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 a form of 11 that consists of the conjugate of the denominator over itself.\newlineThe expression becomes (9×(−5−2))/((−5+2)×(−5−2))(9 \times (-5 - \sqrt{2})) / ((-5 + \sqrt{2}) \times (-5 - \sqrt{2})).
  3. Simplify Numerator Distribution: Simplify the numerator by distributing the multiplication.\newlineSimplify 9×(−5−2)9 \times (-5 - \sqrt{2}):\newline=9×(−5)+9×(−2)= 9 \times (-5) + 9 \times (-\sqrt{2})\newline=−45−92= -45 - 9\sqrt{2}
  4. Simplify Denominator: Simplify the denominator using the difference of squares formula.\newlineSimplify (−5+2)∗(−5−2)(-5 + \sqrt{2}) * (-5 - \sqrt{2}):\newline= (−5)2−(2)2(-5)^2 - (\sqrt{2})^2\newline= 25−225 - 2\newline= 2323
  5. Write Simplified Expression: Write the simplified expression.\newlineThe simplified expression is (−45−92)/23(-45 - 9\sqrt{2}) / 23.\newlineThis fraction is already in simplest form.

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