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Simplify. Rationalize the denominator. \newline3−9−2\frac{3}{-9 - \sqrt{2}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline3−9−2\frac{3}{-9 - \sqrt{2}}
  1. Find Conjugate: Select the conjugate of −9−2-9 - \sqrt{2}.\newlineConjugate of a−ba - \sqrt{b}: a+ba + \sqrt{b}\newlineConjugate of −9−2-9 - \sqrt{2}: −9+2-9 + \sqrt{2}
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it.\newline3−9−2×−9+2−9+2\frac{3}{-9 - \sqrt{2}} \times \frac{-9 + \sqrt{2}}{-9 + \sqrt{2}}
  3. Simplify Numerator: Simplify the numerator by distributing the 33 across the conjugate −9+2-9 + \sqrt{2}.3×(−9+2)=3×(−9)+3×(2)=−27+323 \times (-9 + \sqrt{2}) = 3 \times (-9) + 3 \times (\sqrt{2}) = -27 + 3\sqrt{2}
  4. Simplify Denominator: Simplify the denominator by using the difference of squares formula (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2.(−9−2)∗(−9+2)(-9 - \sqrt{2}) * (-9 + \sqrt{2})=(−9)2−(2)2= (-9)^2 - (\sqrt{2})^2=81−2= 81 - 2=79= 79
  5. Combine Numerator and Denominator: Combine the simplified numerator and denominator. \newline(−27+32)/79(-27 + 3\sqrt{2})/79\newlineThis fraction is already in simplest form.

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