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Simplify. Rationalize the denominator. \newline10−5+2\frac{10}{-5 + \sqrt{2}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline10−5+2\frac{10}{-5 + \sqrt{2}}
  1. Select Conjugate: Select the conjugate of −5+2-5 + \sqrt{2}.\newlineConjugate of a−ba - \sqrt{b}: a+ba + \sqrt{b}\newlineConjugate of −5+2-5 + \sqrt{2}: −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 expression by (-5 - \sqrt{2})/(-5 - \sqrt{2})\(\newline\).\newline(10/(−5+2))×((−5−2)/(−5−2))(10/(-5 + \sqrt{2})) \times ((-5 - \sqrt{2})/(-5 - \sqrt{2}))
  3. Simplify Numerator: Simplify the numerator by distributing the multiplication.\newline10×(−5−2)10 \times (-5 - \sqrt{2})\newline= 10×(−5)+10×(−2)10 \times (-5) + 10 \times (-\sqrt{2})\newline= −50−102-50 - 10\sqrt{2}
  4. Simplify Denominator: Simplify the denominator using the difference of squares formula.\newline(−5+2)×(−5−2)(-5 + \sqrt{2}) \times (-5 - \sqrt{2})\newline=(−5)2−(2)2= (-5)^2 - (\sqrt{2})^2\newline=25−2= 25 - 2\newline=23= 23
  5. Write Simplified Expression: Write the simplified expression.\newline(−50−102)/23(-50 - 10\sqrt{2})/23\newlineThis fraction is already in simplest form.

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