We will apply the "FOIL" method for multiplying two binomials. In this method you will:
- Multiply the First terms of each binomial
- Multiply the Outside terms of each binomial
- Multiply the Inside terms of each binomial
- Multiply the Last terms of each binomial
- Add all of the new terms you have just found together
The mathematical representation of the FOIL method is shown as:
[tex](a + b)(c + d) = ac + ad + bc + bd[/tex]
Let's apply the FOIL method.
- The first two terms of each binomial are [tex]\frac{4}{7}[/tex] and [tex]\frac{8}{7}[/tex]. Multiplied together, we find the new term of [tex]\frac{4}{7} \cdot \frac{8}{7} = \frac{4 \cdot 8}{7 \cdot 7} = \frac{32}{49}[/tex].
- The outside two terms are [tex]\frac{4}{7}[/tex] and [tex]\frac{7}{5} i[/tex]. Multiplied together, these two terms are [tex]\frac{4}{7} \cdot \frac{7}{5}i = \frac{4 \cdot 7}{7 \cdot 5}i = \frac{28}{35}i[/tex].
- The inside two terms are [tex]-i[/tex] and [tex]\frac{8}{7}[/tex]. Multiplied together, these two terms are [tex]- \frac{8}{7} i[/tex].
- The last two terms are [tex]-i[/tex] and [tex]\frac{7}{5} i[/tex]. Multiplied together, these two terms are [tex]-i \cdot \frac{7}{5}i = -\frac{7}{5} i^2[/tex]. Remember though that [tex]i^2 = 1[/tex], so our new term is actually [tex]\frac{7}{5}[/tex].
- Added together, we get: [tex]\frac{32}{49} + \frac{28}{35} i - \frac{8}{7} i + \frac{7}{5} = \frac{160}{245} + \frac{196}{245}i - \frac{280}{245}i + \frac{343}{245} = \frac{503}{245} - \frac{84}{245} i = \frac{503}{245} - \frac{12}{35}i[/tex].
Our final answer is [tex]\boxed{\frac{503}{245} - \frac{12}{35}i}[/tex].
