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

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

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Full solution

Q. Multiply and simplify the following complex numbers:\newline(−1−5i)⋅(1−2i) (-1-5 i) \cdot(1-2 i)
  1. Apply distributive property: Apply the distributive property (also known as the FOIL method for binomials) to multiply the complex numbers.\newline(−1−5i)∗(1−2i)=(−1)∗(1)+(−1)∗(−2i)+(−5i)∗(1)+(−5i)∗(−2i)(-1-5i)*(1-2i) = (-1)*(1) + (-1)*(-2i) + (-5i)*(1) + (-5i)*(-2i)
  2. Perform multiplication for each term: Perform the multiplication for each term.\newline(−1)×(1)=−1(-1)\times(1) = -1\newline(−1)×(−2i)=2i(-1)\times(-2i) = 2i\newline(−5i)×(1)=−5i(-5i)\times(1) = -5i\newline(−5i)×(−2i)=10i2(-5i)\times(-2i) = 10i^2
  3. Replace i2i^2 with −1-1: Remember that i2=−1i^2 = -1, so replace i2i^2 with −1-1 in the last term.\newline10i2=10(−1)=−1010i^2 = 10(-1) = -10
  4. Combine like terms: Combine like terms (real with real and imaginary with imaginary).\newline(−1)+(2i)+(−5i)+(−10)(-1) + (2i) + (-5i) + (-10)
  5. Simplify expression: Simplify the expression by adding/subtracting the real parts and the imaginary parts.\newlineReal: −1−10=−11-1 - 10 = -11\newlineImaginary: 2i−5i=−3i2i - 5i = -3i
  6. Write final answer: Write the final answer as a complex number in the form a+bia + bi.\newline−11−3i-11 - 3i

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