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

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

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Full solution

Q. Multiply and simplify the following complex numbers:\newline(−4+2i)⋅(4−4i) (-4+2 i) \cdot(4-4 i)
  1. Apply distributive property: Apply the distributive property (also known as the FOIL method for binomials) to multiply the two complex numbers.\newline(−4+2i)∗(4−4i)=(−4⋅4)+(−4⋅−4i)+(2i⋅4)+(2i⋅−4i)(-4+2i)*(4-4i) = (-4\cdot 4) + (-4\cdot -4i) + (2i\cdot 4) + (2i\cdot -4i)
  2. Perform multiplication for each term: Perform the multiplication for each term.\newline(−4×4)=−16(-4 \times 4) = -16\newline(−4×−4i)=16i(-4 \times -4i) = 16i\newline(2i×4)=8i(2i \times 4) = 8i\newline(2i×−4i)=−8i2(2i \times -4i) = -8i^2
  3. Combine like terms and simplify: Combine like terms and remember that i2=−1i^2 = -1.\newline−16+16i+8i−8(−1)=−16+24i+8-16 + 16i + 8i - 8(-1) = -16 + 24i + 8
  4. Add real and imaginary parts separately: Simplify the expression by adding the real parts and the imaginary parts separately.\newline−16+8=−8-16 + 8 = -8 (real part)\newline24i24i (imaginary part)
  5. Write final answer as complex number: Write the final answer as a complex number.\newlineThe product of the complex numbers (−4+2i)(-4+2i) and (4−4i)(4-4i) is −8+24i-8 + 24i.

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