Question

Simplify the expression to a + bi form:
left parenthesis, minus, 9, plus, 5, i, right parenthesis, left parenthesis, minus, 5, minus, 7, i, right parenthesis
(−9+5i)(−5−7i)

Answer 1

To simplify the expression \((−9 + 5i)(−5 − 7i)\), we can use the distributive property (also known as the FOIL method for binomials).

1. Multiply the first terms:
\[
(-9)(-5) = 45
\]

2. Multiply the outer terms:
\[
(-9)(-7i) = 63i
\]

3. Multiply the inner terms:
\[
(5i)(-5) = -25i
\]

4. Multiply the last terms:
\[
(5i)(-7i) = -35i^2
\]

Since \(i^2 = -1\), we can replace \(-35i^2\) with \(35\):
\[
-35i^2 = -35(-1) = 35
\]

Now, combine all the results:
\[
45 + 63i - 25i + 35
\]

Combine the real parts (45 and 35) and the imaginary parts (63i and -25i):
\[
(45 + 35) + (63i - 25i) = 80 + 38i
\]

So, the expression simplified to \(a + bi\) form is:
\[
\boxed{80 + 38i}
\]