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Express as a complex number in simplest a+bi form: (-5-20 i)/(-6-7i) Answer:

Express as a complex number in simplest a+bi form:\newline−5−20i−6−7i \frac{-5-20 i}{-6-7 i} \newlineAnswer:

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Full solution

Q. Express as a complex number in simplest a+bi form:\newline−5−20i−6−7i \frac{-5-20 i}{-6-7 i} \newlineAnswer:
  1. Multiply by Conjugate: To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number a+bia + bi is a−bia - bi. So, the conjugate of −6−7i-6 - 7i is −6+7i-6 + 7i.
  2. Numerator Multiplication: Now, we multiply both the numerator and the denominator by the conjugate of the denominator: (−5−20i)×(−6+7i)(−6−7i)×(−6+7i)\frac{(-5 - 20i) \times (-6 + 7i)}{(-6 - 7i) \times (-6 + 7i)}.
  3. Combine Like Terms (Numerator): First, we'll multiply out the numerator:\newline(−5×−6)+(−5×7i)+(−20i×−6)+(−20i×7i)(-5 \times -6) + (-5 \times 7i) + (-20i \times -6) + (-20i \times 7i).\newlineThis simplifies to:\newline(30)+(35i)+(120i)+(−140)(30) + (35i) + (120i) + (-140).
  4. Denominator Multiplication: Now, we'll combine like terms in the numerator:\newline30−140+35i+120i30 - 140 + 35i + 120i simplifies to:\newline−110+155i-110 + 155i.
  5. Combine Like Terms (Denominator): Next, we'll multiply out the denominator:\newline(−6×−6)+(−6×7i)+(−7i×−6)+(−7i×7i)(-6 \times -6) + (-6 \times 7i) + (-7i \times -6) + (-7i \times 7i).\newlineThis simplifies to:\newline(36)−(42i)+(42i)−(49)(36) - (42i) + (42i) - (49).
  6. Final Complex Number: Now, we'll combine like terms in the denominator:\newline36−49−42i+42i36 - 49 - 42i + 42i simplifies to:\newline−13-13, since the imaginary parts cancel each other out.
  7. Simplify Real and Imaginary Parts: We now have the complex number in the form of (−110+155i)/−13(-110 + 155i) / -13. To simplify, we divide both the real part and the imaginary part by −13-13: (−110/−13)+(155i/−13)(-110 / -13) + (155i / -13).
  8. Round or Express as Fraction: Simplifying both parts gives us: 8.46153846154−11.9230769231i8.46153846154 - 11.9230769231i. However, we should round to a reasonable number of decimal places or write as a fraction if possible.
  9. Round or Express as Fraction: Simplifying both parts gives us: 8.46153846154−11.9230769231i8.46153846154 - 11.9230769231i. However, we should round to a reasonable number of decimal places or write as a fraction if possible.Rounding to two decimal places, we get: 8.46−11.92i8.46 - 11.92i. Alternatively, we can express the fractions in their simplest form: (110/13)−(155/13)i(110/13) - (155/13)i.

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