Divide the following complex numbers \$ (8 + 3i) / (2 + 6i)\$.

The given complex number

\$ (8 + 3i) / (2 + 6i)\$

Find the conjugate of the denominator:

Conjugate of (2 + 6i) = 2 - 6i

Multiply the numerator and the denominator by the conjugate of the denominator:

\$ (8 + 3i) / (2 + 6i) times ( 2 - 6i) / (2 -6i)\$

Since \$i^2 = - 1\$.

Calculate the numerator and plug the \$i^2 = -1\$ and then make it simplify:

\$ (2 + 6i)(2 - 6i) = 2^2 - (6i)^2\$

\$(a - b) (a + b) = a^2 - b^2\$

\$ 4 - 36i^2 = 4 - 36(-1) = 4 + 36 = 40\$

Calculate the numerator and plug the \$i^2 = -1\$ and then make it simplify:

\$ (8 + 3i)(2 - 6i)\$

⇒ \$8 times 2 + 8 times (- 6i) + 3i times 2 + 3i times (- 6i)\$

⇒ \$ 16 - 48i + 6i -18i^2\$

⇒ \$ 16 - 42i - 18(-1)\$

⇒ 16 - 42i + 18

⇒ 34 - 42i

Combine the results and simplify:

\$ (34 - 42i) / (40)\$

\$ (34) / (40) - (42)/(40) i\$

⇒ 0.85 - 1.05i

So, the result of the division in standard form is \$ (8 + 3i) / (2 + 6i) = 0.85 - 1.05i\$.

It matches our calculation. So, this choice is correct

0.85 - 1.05i

(1.05−0.85i) is incorrect due to swapped parts and wrong magnitudes. The correct answer has real part 0.85 and imaginary part −1.05i

1.05 - 0.85i

(0.85 + 1.05i) has the wrong sign for the imaginary part. The correct result has a negative imaginary part, not positive

0.85 + 1.05i

(1.05 + 0.85i) is wrong because the real and imaginary parts are reversed in sign and magnitude

1.05 + 0.85i

Option

A

Can you summarize what you’ve understood in the above steps?