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

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Full solution

Q. Simplify. Rationalize the denominator.\newline3−2+2\frac{3}{-2 + \sqrt{2}}
  1. Select Conjugate: Select the conjugate of −2+2-2 + \sqrt{2}.\newlineConjugate of a−ba - \sqrt{b}: a+ba + \sqrt{b}\newlineConjugate of −2+2-2 + \sqrt{2}: −2−2-2 - \sqrt{2}
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it.\newline3−2+2⋅−2−2−2−2\frac{3}{-2 + \sqrt{2}} \cdot \frac{-2 - \sqrt{2}}{-2 - \sqrt{2}}
  3. Simplify Numerator: Simplify the numerator by distributing 33 to both terms in the conjugate.\newline3×(−2)−3×(2)3 \times (-2) - 3 \times (\sqrt{2})\newline= −6−32-6 - 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.(−2+2)∗(−2−2)(-2 + \sqrt{2}) * (-2 - \sqrt{2})=(−2)2−(2)2= (-2)^2 - (\sqrt{2})^2=4−2= 4 - 2=2= 2
  5. Combine Numerator and Denominator: Combine the simplified numerator and denominator.\newline(−6−32)/2(-6 - 3\sqrt{2})/2\newlineThis fraction is already in simplest form.

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