Questions: Perform the indicated operations. Write the answer in standard form, (a+bi). [ (7+6 i)(-4-5 i)= ]

Transcript text: Perform the indicated operations. Write the answer in standard form, $a+b i$. \[ (7+6 i)(-4-5 i)= \]

Solution

Solution Steps

Step 1: Multiplication of Two Complex Numbers

To multiply two complex numbers $(a + bi)(c + di)$, we use the distributive property to expand the product: \[ (a + bi)(c + di) = ac + adi + bci + bdi^2 \] Since $i^2 = -1$, we simplify the expression to get $ac - bd + (ad + bc)i$. Substituting the given values, we get $7 \times -4 - 6 \times -5 + (7 \times -5 + 6 \times -4)i$.

Step 2: Simplification

After simplification, the real part is $2$ and the imaginary part is $-59$.