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Simplify. Rationalize the denominator. \newline4−4−2\frac{4}{-4 - \sqrt{2}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline4−4−2\frac{4}{-4 - \sqrt{2}}
  1. Multiply by Conjugate: We multiply the numerator and the denominator by the conjugate of the denominator to rationalize it.\newline(4×(−4+2))/((−4−2)×(−4+2))(4 \times (-4 + \sqrt{2})) / ((-4 - \sqrt{2}) \times (-4 + \sqrt{2}))
  2. Simplify Numerator: Now we simplify the numerator by distributing the 44 across the conjugate.4×(−4)+4×2=−16+424 \times (-4) + 4 \times \sqrt{2} = -16 + 4\sqrt{2}
  3. Simplify Denominator: Next, we simplify the denominator by using the difference of squares formula, which states that (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2.(−4)2−(2)2=16−2=14(-4)^2 - (\sqrt{2})^2 = 16 - 2 = 14
  4. Write Simplified Expression: Now we write the simplified expression with the rationalized denominator. (−16+42)/14(-16 + 4\sqrt{2}) / 14
  5. Further Simplify Expression: We can simplify the expression further by dividing both terms in the numerator by the denominator.(−1614)+(4214)(-\frac{16}{14}) + (\frac{4\sqrt{2}}{14})
  6. Reduce Fractions: Simplify the fractions by reducing them to their simplest form. \newline−87+227-\frac{8}{7} + \frac{2\sqrt{2}}{7}

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