Question

Perform the indicated operation and write the result in the form a+bi:

(4/7 - i)(8/7 + 7/5i)

Answer 1

We will apply the "FOIL" method for multiplying two binomials. In this method you will:

1. Multiply the First terms of each binomial
2. Multiply the Outside terms of each binomial
3. Multiply the Inside terms of each binomial
4. Multiply the Last terms of each binomial
5. Add all of the new terms you have just found together

The mathematical representation of the FOIL method is shown as:

`(a + b)(c + d) = ac + ad + bc + bd`

Let's apply the FOIL method.

1. The first two terms of each binomial are
   `(4)/(7)` and
   `(8)/(7)`. Multiplied together, we find the new term of
   `(4)/(7) \cdot (8)/(7) = (4 \cdot 8)/(7 \cdot 7) = (32)/(49)`.
2. The outside two terms are
   `(4)/(7)` and
   `(7)/(5) i`. Multiplied together, these two terms are
   `(4)/(7) \cdot (7)/(5)i = (4 \cdot 7)/(7 \cdot 5)i = (28)/(35)i`.
3. The inside two terms are
   `-i` and
   `(8)/(7)`. Multiplied together, these two terms are
   `- (8)/(7) i`.
4. The last two terms are
   `-i` and
   `(7)/(5) i`. Multiplied together, these two terms are
   `-i \cdot (7)/(5)i = -(7)/(5) i^2`. Remember though that
   `i^2 = 1`, so our new term is actually
   `(7)/(5)`.
5. Added together, we get:
   `(32)/(49) + (28)/(35) i - (8)/(7) i + (7)/(5) = (160)/(245) + (196)/(245)i - (280)/(245)i + (343)/(245) = (503)/(245) - (84)/(245) i = (503)/(245) - (12)/(35)i`.

Our final answer is
`\boxed{(503)/(245) - (12)/(35)i}`.