Questions: (4-5 i)(1-i)-(3-i)(3+i) (4-5 i)(1-i)-(3-i)(3+i)= (Simplify your answer. Type your answer in the form a + bi.)

Solution Steps

To solve the given expression, we need to perform operations on complex numbers. First, expand each product using the distributive property (also known as FOIL for binomials). Then, simplify the resulting expression by combining like terms. Remember that \(i^2 = -1\).

We start with the expression: \[ (4 - 5i)(1 - i) - (3 - i)(3 + i) \] First, we expand each product using the distributive property.

For the first product: \[ (4 - 5i)(1 - i) = 4 \cdot 1 + 4 \cdot (-i) - 5i \cdot 1 - 5i \cdot (-i) = 4 - 4i - 5i + 5i^2 \] Since \(i^2 = -1\), we have: \[ 5i^2 = 5(-1) = -5 \] Thus, the first product simplifies to: \[ 4 - 4i - 5i - 5 = -1 - 9i \]

For the second product: \[ (3 - i)(3 + i) = 3 \cdot 3 + 3 \cdot i - i \cdot 3 - i \cdot i = 9 + 3i - 3i - i^2 \] Again, using \(i^2 = -1\): \[ -i^2 = -(-1) = 1 \] So, the second product simplifies to: \[ 9 + 1 = 10 \]

Now we substitute back into the original expression: \[ (-1 - 9i) - 10 \] This simplifies to: \[ -1 - 10 - 9i = -11 - 9i \]

Final Answer