Questions: Perform the indicated operations. (5-2 i)(8+i) (5-2 i)(8+i)= (Simplify your answer. Type your answer in the form a + bi.

Solution Steps

To solve the given problem, we need to perform the multiplication of two complex numbers. We will use the distributive property (FOIL method) to expand the expression and then combine like terms.

To multiply the complex numbers \((5 - 2i)\) and \((8 + i)\), we use the distributive property (FOIL method):

\[ (5 - 2i)(8 + i) = 5 \cdot 8 + 5 \cdot i - 2i \cdot 8 - 2i \cdot i \]

Calculate each term separately:

\[ 5 \cdot 8 = 40 \] \[ 5 \cdot i = 5i \] \[ -2i \cdot 8 = -16i \] \[ -2i \cdot i = -2i^2 \]

Combine the real and imaginary parts:

\[ 40 + 5i - 16i - 2i^2 \]

Since \(i^2 = -1\):

\[ -2i^2 = -2(-1) = 2 \]

So the expression becomes:

\[ 40 + 5i - 16i + 2 \]

Combine the real parts and the imaginary parts:

\[ (40 + 2) + (5i - 16i) = 42 - 11i \]

Final Answer