Answer:
[tex]s = \frac{5i}{-2-6i} = -\frac{30}{40}-\frac{10}{40}i[/tex]
Step-by-step explanation:
We have the following complex number [tex]s = \frac{5i}{-2-6i}[/tex], we proceed to simplify the expression as follows:
1) [tex]\frac{5i}{-2-6i}[/tex] Given.
2) [tex](5i)\cdot (-2-6i)^{-1}[/tex] Definition of division.
3) [tex][(5i)\cdot (-2-6i)^{-1}]\cdot [(-2+6i)\cdot (-2+6i)^{-1}][/tex] Modulative and associative properties/Existence of the additive inverser
4) [tex][(5i)\cdot (-2+6i)]\cdot [(-2-6i)^{-1}\cdot (-2+6i)^{-1}][/tex] Commutative and associative properties.
5) [tex][(5i)\cdot (-2+6i)]\cdot [(-2-6i)\cdot (-2+6i)]^{-1}[/tex] [tex]a^{c}\cdot b^{c} = (a\cdot b)^{c}[/tex]
6) [tex][(5i)\cdot (-2+6i)]\cdot [4+36]^{-1}[/tex] [tex](a+b)\cdot (a-b) = a^{2}-b^{2}[/tex]/Definition of complex number/ [tex]a\cdot (-b) = -a\cdot b[/tex]
7) [tex][(5i)\cdot (-2)+(5i)\cdot (6i)]\cdot 40^{-1}[/tex] Definition of sum.
8) [tex](-10\cdot i+30\cdot i^{2})\cdot 40^{-1}[/tex] [tex]a\cdot (-b) = -a\cdot b[/tex]/Associative and commutative properties.
9) [tex](-30-10i)\cdot 40^{-1}[/tex] Commutative properties/Definition of complex number/ [tex]a\cdot (-b) = -a\cdot b[/tex]
10) [tex]-30\cdot 40^{-1}-(10i)\cdot 40^{-1}[/tex] Distributive property.
11) [tex]-\frac{30}{40}-\frac{10}{40}i[/tex] Definition of division/Result.
[tex]s = \frac{5i}{-2-6i} = -\frac{30}{40}-\frac{10}{40}i[/tex]
