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Simplify. Rationalize the denominator. \newline9−8−2\frac{9}{-8 - \sqrt{2}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline9−8−2\frac{9}{-8 - \sqrt{2}}
  1. Find Conjugate: Select the conjugate of −8−2-8 - \sqrt{2}.\newlineConjugate of a number in the form a−ba - \sqrt{b} is a+ba + \sqrt{b}.\newlineTherefore, the conjugate of −8−2-8 - \sqrt{2} is −8+2-8 + \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 −8+2−8+2\frac{-8 + \sqrt{2}}{-8 + \sqrt{2}}.\newline9−8−2×−8+2−8+2\frac{9}{-8 - \sqrt{2}} \times \frac{-8 + \sqrt{2}}{-8 + \sqrt{2}}
  3. Simplify Numerator: Simplify the numerator.\newlineMultiply 99 by the conjugate −8+2-8 + \sqrt{2}.\newline9×(−8)+9×29 \times (-8) + 9 \times \sqrt{2}\newline−72+92-72 + 9\sqrt{2}
  4. Simplify Denominator: Simplify the denominator using the difference of squares formula.\newline(−8−2)×(−8+2)=(−8)2−(2)2(-8 - \sqrt{2}) \times (-8 + \sqrt{2}) = (-8)^2 - (\sqrt{2})^2\newline64−264 - 2\newline6262
  5. Write Simplified Expression: Write the simplified expression.\newline−72+9262\frac{-72 + 9\sqrt{2}}{62}\newlineThis fraction is already in simplest form.

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