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Multiply and simplify the following complex numbers: (1-4i)*(-4+2i)

Multiply and simplify the following complex numbers:\newline(1−4i)⋅(−4+2i) (1-4 i) \cdot(-4+2 i)

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Full solution

Q. Multiply and simplify the following complex numbers:\newline(1−4i)⋅(−4+2i) (1-4 i) \cdot(-4+2 i)
  1. Distribute terms: Distribute each term in the first complex number by each term in the second complex number.\newline(1−4i)(−4+2i)=1(−4)+1(2i)+(−4i)(−4)+(−4i)(2i)(1-4i)(-4+2i) = 1(-4) + 1(2i) + (-4i)(-4) + (-4i)(2i)
  2. Perform multiplication: Perform the multiplication for each term.\newline1⋅(−4)=−41 \cdot (-4) = -4\newline1⋅(2i)=2i1 \cdot (2i) = 2i\newline(−4i)⋅(−4)=16i(-4i) \cdot (-4) = 16i\newline(−4i)⋅(2i)=−8i2(-4i) \cdot (2i) = -8i^2
  3. Combine like terms: Combine like terms and remember that i2=−1i^2 = -1.\newline−4+2i+16i−8(−1)=−4+18i+8-4 + 2i + 16i - 8(-1) = -4 + 18i + 8
  4. Add real and imaginary parts: Add the real parts and the imaginary parts.\newline(−4+8)+(2i+16i)=4+18i(-4 + 8) + (2i + 16i) = 4 + 18i

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