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Multiply and simplify the following complex numbers: (1-5i)*(3-2i)

Multiply and simplify the following complex numbers:\newline(1−5i)⋅(3−2i) (1-5 i) \cdot(3-2 i)

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Full solution

Q. Multiply and simplify the following complex numbers:\newline(1−5i)⋅(3−2i) (1-5 i) \cdot(3-2 i)
  1. Distribute terms: Distribute each term in the first complex number by each term in the second complex number.\newline(1−5i)∗(3−2i)=1∗(3−2i)−5i∗(3−2i)(1-5i)*(3-2i) = 1*(3-2i) - 5i*(3-2i)
  2. Multiply real and imaginary parts: Multiply the real parts and the imaginary parts separately.\newline1×3−1×2i−5i×3+5i×2i1\times3 - 1\times2i - 5i\times3 + 5i\times2i
  3. Combine like terms: Combine like terms and remember that i2=−1i^2 = -1.\newline1×3=31 \times 3 = 3\newline−1×2i=−2i-1 \times 2i = -2i\newline−5i×3=−15i-5i \times 3 = -15i\newline5i×2i=10i2=10×(−1)=−105i \times 2i = 10i^2 = 10 \times (-1) = -10 (since i2=−1i^2 = -1)
  4. Add real and imaginary parts: Add the real parts and the imaginary parts together. 3−2i−15i−103 - 2i - 15i - 10
  5. Combine real and imaginary numbers: Combine the real numbers and the imaginary numbers. (3−10)+(−2i−15i)(3 - 10) + (-2i - 15i)
  6. Simplify expression: Simplify the expression by adding the real parts and the imaginary parts.\newline3−10=−73 - 10 = -7\newline−2i−15i=−17i-2i - 15i = -17i
  7. Write final answer: Write the final answer in the form of a complex number (a+bi)(a + bi).\newline−7−17i-7 - 17i

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