To find [tex]\((\sqrt{2} - \sqrt{2}i)^8\)[/tex] and write the answer in rectangular form, we can follow these steps:
1. Express the complex number in polar form:
[tex]\[
z = \sqrt{2} - \sqrt{2}i
\][/tex]
We write [tex]\(z\)[/tex] in the polar form [tex]\(re^{i\theta}\)[/tex] where [tex]\(r\)[/tex] is the modulus and [tex]\(\theta\)[/tex] is the argument (angle).
2. Calculate the modulus [tex]\(r\)[/tex]:
[tex]\[
r = \sqrt{(\sqrt{2})^2 + (-\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2
\][/tex]
3. Find the argument [tex]\(\theta\)[/tex]:
The argument [tex]\(\theta\)[/tex] can be found using the arctangent function:
[tex]\[
\theta = \arctan\left(\frac{-\sqrt{2}}{\sqrt{2}}\right) = \arctan(-1) = -\frac{\pi}{4}
\][/tex]
Therefore, in polar form:
[tex]\[
z = 2e^{-i\pi/4}
\][/tex]
4. Use De Moivre's Theorem to raise [tex]\(z\)[/tex] to the 8th power:
According to De Moivre's Theorem:
[tex]\[
z^n = (re^{i\theta})^n = r^n e^{in\theta}
\][/tex]
Applying this to our problem:
[tex]\[
z^8 = \left(2e^{-i\pi/4}\right)^8 = 2^8 e^{-i 8 \cdot \pi/4}
\][/tex]
Simplify the exponent:
[tex]\[
e^{-i 8 \pi/4} = e^{-i 2 \pi} = e^{0} = 1
\][/tex]
Thus,
[tex]\[
z^8 = 2^8 \cdot 1 = 2^8 = 256
\][/tex]
5. Convert back to rectangular form:
The result can be written as:
[tex]\[
256 + 0i
\][/tex]
Therefore, the result of [tex]\((\sqrt{2} - \sqrt{2}i)^8\)[/tex] in rectangular form is:
[tex]\[
\boxed{256 + 0i}
\][/tex]
Or simplified:
[tex]\[
\boxed{256}
\][/tex]
