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Multiply and simplify the following complex numbers: (1+5i)*(-3-i)

Multiply and simplify the following complex numbers:\newline(1+5i)⋅(−3−i) (1+5 i) \cdot(-3-i)

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Multiply numbers written in scientific notationright chevron icon

Full solution

Q. Multiply and simplify the following complex numbers:\newline(1+5i)⋅(−3−i) (1+5 i) \cdot(-3-i)
  1. Distribute terms: Distribute each term of the first complex number by each term of the second complex number.\newline(1+5i)(−3−i)=1(−3)+1(−i)+5i(−3)+5i(−i)(1+5i)(-3-i) = 1(-3) + 1(-i) + 5i(-3) + 5i(-i)
  2. Multiply terms: Multiply the terms.\newline1⋅(−3)=−31 \cdot (-3) = -3\newline1⋅(−i)=−i1 \cdot (-i) = -i\newline5i⋅(−3)=−15i5i \cdot (-3) = -15i\newline5i⋅(−i)=5i25i \cdot (-i) = 5i^2\newlineSince i2=−1i^2 = -1, we replace 5i25i^2 with 5⋅(−1)5 \cdot (-1).
  3. Replace i2i^2: Combine like terms and simplify.\newline−3−i−15i+5∗(−1)-3 - i - 15i + 5*(-1)\newlineCombine the real parts: −3+5∗(−1)=−3−5=−8-3 + 5*(-1) = -3 - 5 = -8\newlineCombine the imaginary parts: −i−15i=−16i-i - 15i = -16i\newlineSo, the simplified form is −8−16i-8 - 16i.

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