Perform the complex number division below:

5 - 7i | |

7 + 3i |

Complex number division involves multiplying numerator and denominator by the

If the denominator is c + di, the conjugate is c - di. Multiplying top and bottom by the conjugate (7 - 3i), we get:

(5 - 7i)(7 - 3i) | |

(7 + 3i)(7 - 3i) |

The formula for this using the FOIL method is: (a * c) + (b * c) + (a * d) + (b * d) where:

a = 7, b = 3, c = 7, and d = -3

(7 + 3i)(7 - 3i) = (7 * 7) + (3i * 7) + (7 * -3i) + (3i * -3i)

(7 + 3i)(7 - 3i) = 49 + 21i - 21i - 9i

(7 + 3i)(7 - 3i) = 49 + (21 - 21)i - 9i

(7 + 3i)(7 - 3i) = 49 - 9i

i

(7 + 3i)(7 - 3i) = 49 - 9* (-1)

(7 + 3i)(7 - 3i) = 49 + 9

(7 + 3i)(7 - 3i) = (49 + 9)

(7 + 3i)(7 - 3i) =

The formula for this using the FOIL method is: (a * c) + (b * c) + (a * d) + (b * d) where:

a = 5, b = -7, c = 7, and d = -3

(5 - 7i)(7 - 3i) = (5 * 7) + (-7i * 7) + (5 * -3i) + (-7i * -3i)

(5 - 7i)(7 - 3i) = 35 - 49i - 15i + 21i

(5 - 7i)(7 - 3i) = 35 + (-49 - 15)i + 21i

(5 - 7i)(7 - 3i) = 35 - 64i + 21i

i

(5 - 7i)(7 - 3i) = 35 - 64i + 21* (-1)

(5 - 7i)(7 - 3i) = 35 - 64i - 21

(5 - 7i)(7 - 3i) = (35 - 21) - 64i

(5 - 7i)(7 - 3i) =

5 - 7i | |

7 + 3i |

= |

14 - 64i |

58 |

Our fraction is not fully reduced. The Greatest Common Factor (GCF) of 14, -64, and 58 is 2. Reducing our fraction by the GCF, we get our answer:

5 - 7i |

7 + 3i |

= |

7 - 32i |

29 |

5 - 7i | |

7 + 3i |

= |

7 - 32i |

29 |