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Simplify. Rationalize the denominator. \newline5−10+2\frac{5}{-10 + \sqrt{2}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline5−10+2\frac{5}{-10 + \sqrt{2}}
  1. Identify Conjugate: Identify the conjugate of the denominator −10+2-10 + \sqrt{2}.\newlineThe conjugate of a number of the form a+ba + \sqrt{b} is a−ba - \sqrt{b}. Therefore, the conjugate of −10+2-10 + \sqrt{2} is −10−2-10 - \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 (−10−2)/(−10−2)(-10 - \sqrt{2})/(-10 - \sqrt{2}) which is equal to 11 and thus does not change the value of the expression.\newline(5⋅(−10−2))/((−10+2)⋅(−10−2))(5 \cdot (-10 - \sqrt{2})) / ((-10 + \sqrt{2}) \cdot (-10 - \sqrt{2}))
  3. Simplify Numerator: Simplify the numerator by distributing the multiplication.\newline5×(−10)+5×(−2)5 \times (-10) + 5 \times (-\sqrt{2})\newline= −50−5×2-50 - 5 \times \sqrt{2}
  4. Simplify Denominator: Simplify the denominator using the difference of squares formula.\newline(−10)2−(2)2(-10)^2 - (\sqrt{2})^2\newline= 100−2100 - 2\newline= 9898
  5. Write Simplified Expression: Write the simplified expression with the rationalized denominator.\newline(−50−52)/98(-50 - 5 \sqrt{2}) / 98\newlineThis fraction is already in simplest form.

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