To simplify [tex]\(\frac{z_2}{z_1}\)[/tex] where [tex]\(z_1 = 1 + i\)[/tex] and [tex]\(z_2 = 2 - 2\sqrt{3}i\)[/tex], follow these steps:
1. Write down the complex numbers:
Let [tex]\(z_1 = 1 + i\)[/tex] and [tex]\(z_2 = 2 - 2\sqrt{3}i\)[/tex].
2. Understand the division of complex numbers:
To divide two complex numbers, we can multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(z_1 = 1 + i\)[/tex] is [tex]\(1 - i\)[/tex].
3. Set up the division:
[tex]\[
\frac{z_2}{z_1} = \frac{2 - 2\sqrt{3}i}{1 + i} \cdot \frac{1 - i}{1 - i}
\][/tex]
4. Multiply the numerators and the denominators:
[tex]\[
\text{Numerator: } (2 - 2\sqrt{3}i)(1 - i)
\][/tex]
Using the distributive property:
[tex]\[
(2 - 2\sqrt{3}i)(1 - i) = 2 \cdot 1 + 2 \cdot (-i) - 2\sqrt{3}i \cdot 1 - 2\sqrt{3}i \cdot (-i)
\][/tex]
Simplify each term:
[tex]\[
= 2 - 2i - 2\sqrt{3}i + 2\sqrt{3}i^2
= 2 - 2i - 2\sqrt{3}i + 2\sqrt{3}(-1)
\][/tex]
[tex]\[
= 2 - 2i - 2\sqrt{3}i - 2\sqrt{3}
\][/tex]
[tex]\[
= (2 - 2\sqrt{3}) + (-2 - 2\sqrt{3})i
\][/tex]
5. Multiply the denominators:
[tex]\[
\text{Denominator: } (1 + i)(1 - i)
\][/tex]
Using the distributive property:
[tex]\[
= 1 \cdot 1 + 1 \cdot (-i) + i \cdot 1 + i \cdot (-i)
\][/tex]
[tex]\[
= 1 - i + i - i^2
\][/tex]
[tex]\[
= 1 - i^2
= 1 - (-1)
= 1 + 1
= 2
\][/tex]
6. Combine the terms:
[tex]\[
\frac{z_2}{z_1} = \frac{(2 - 2\sqrt{3}) + (-2 - 2\sqrt{3})i}{2}
\][/tex]
Simplify the division:
[tex]\[
= \frac{2 - 2\sqrt{3}}{2} + \frac{(-2 - 2\sqrt{3})i}{2}
\][/tex]
[tex]\[
= 1 - \sqrt{3} - i(1 + \sqrt{3})
\][/tex]
Therefore, the simplified form of [tex]\(\frac{z_2}{z_1}\)[/tex] is:
[tex]\[
1 - \sqrt{3} - i(1 + \sqrt{3})
\][/tex]
The correct answer is:
A. [tex]\(1 - \sqrt{3} - i(1 + \sqrt{3})\)[/tex]
