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Multiply and simplify the following complex numbers: (-4+4i)*(3+2i)

Multiply and simplify the following complex numbers:\newline(−4+4i)⋅(3+2i) (-4+4 i) \cdot(3+2 i)

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Full solution

Q. Multiply and simplify the following complex numbers:\newline(−4+4i)⋅(3+2i) (-4+4 i) \cdot(3+2 i)
  1. Apply distributive property: Apply the distributive property (also known as the FOIL method for binomials) to multiply the complex numbers.\newline(−4+4i)∗(3+2i)=(−4∗3)+(−4∗2i)+(4i∗3)+(4i∗2i)(-4+4i)*(3+2i) = (-4*3) + (-4*2i) + (4i*3) + (4i*2i)
  2. Multiply real and imaginary parts: Multiply the real parts and the imaginary parts separately.\newline(−4×3)=−12(-4 \times 3) = -12 (Real part)\newline(−4×2i)=−8i(-4 \times 2i) = -8i (Imaginary part)\newline(4i×3)=12i(4i \times 3) = 12i (Imaginary part)\newline(4i×2i)=8i2(4i \times 2i) = 8i^2 (Imaginary part squared, where i2=−1i^2 = -1)
  3. Combine like terms: Combine the like terms (real with real and imaginary with imaginary).\newline−12+(−8i)+12i+8i2-12 + (-8i) + 12i + 8i^2\newlineSince i2=−1i^2 = -1, we replace 8i28i^2 with 8(−1)8(-1).\newline−12+(−8i)+12i−8-12 + (-8i) + 12i - 8
  4. Simplify the expression: Simplify the expression by combining like terms.\newline−12−8+(−8i+12i)-12 - 8 + (-8i + 12i)\newline−20+4i-20 + 4i

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