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Simplify. Rationalize the denominator. \newline3−6+2\frac{3}{-6 + \sqrt{2}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline3−6+2\frac{3}{-6 + \sqrt{2}}
  1. Select Conjugate: Select the conjugate of −6+2-6 + \sqrt{2}.\newlineConjugate of a−ba - \sqrt{b}: a+ba + \sqrt{b}\newlineConjugate of −6+2-6 + \sqrt{2}: −6−2-6 - \sqrt{2}
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it.\newline3−6+2×−6−2−6−2\frac{3}{-6 + \sqrt{2}} \times \frac{-6 - \sqrt{2}}{-6 - \sqrt{2}}
  3. Simplify Numerator: Simplify the numerator by distributing the 33 across the conjugate −6−2-6 - \sqrt{2}.3×(−6)−3×(2)=−18−323 \times (-6) - 3 \times (\sqrt{2}) = -18 - 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.\newline(−6+2)∗(−6−2)(-6 + \sqrt{2}) * (-6 - \sqrt{2})\newline=(−6)2−(2)2= (-6)^2 - (\sqrt{2})^2\newline=36−2= 36 - 2\newline=34= 34
  5. Combine Numerator and Denominator: Combine the simplified numerator and denominator. (−18−32)/34(-18 - 3 \sqrt{2})/34
  6. Divide and Simplify: Divide each term in the numerator by the denominator.\newline−1834−3234-\frac{18}{34} - \frac{3 \sqrt{2}}{34}\newlineSimplify the fractions.\newline−917−3234-\frac{9}{17} - \frac{3 \sqrt{2}}{34}

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