Question

Find the product and write the result in standard form.
left parenthesis 8 minus 7 i right parenthesis left parenthesis negative 2 minus 3 i right parenthesis

Answer 1

To find the product of the complex numbers \( (8 - 7i) \) and \( (-2 - 3i) \), we will use the distributive property (also known as the FOIL method for binomials).

\[
(8 - 7i)(-2 - 3i)
\]

Now we distribute each term in the first complex number by each term in the second complex number:

1. First: \( 8 \cdot (-2) = -16 \)
2. Outer: \( 8 \cdot (-3i) = -24i \)
3. Inner: \( -7i \cdot (-2) = 14i \)
4. Last: \( -7i \cdot (-3i) = 21i^2 \)

Since \( i^2 = -1 \), we can rewrite \( 21i^2 \) as \( 21(-1) = -21 \).

Now we can combine all these results:

\[
-16 - 24i + 14i - 21
\]

Combining the real parts and the imaginary parts:

Real part: \( -16 - 21 = -37 \)

Imaginary part: \( -24i + 14i = -10i \)

Putting it all together, we have:

\[
-37 - 10i
\]

Thus, the product in standard form is:

\[
\boxed{-37 - 10i}
\]