Using de Moivre's theorem to compute $$ (-\sqrt{2}-\sqrt{2} i)^{9} $$ we find that the answer is $$ 256 \sqrt{2}-256 \sqrt{2} i $$ $$ 256 \sqrt{2}+256 \sqrt{2} i $$ None of the others $$ -256 \sqrt{2}+256 \sqrt{2} i $$ $$ -256 \sqrt{2}-256 \sqrt{2} i $$

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We start with
$$
z=-\sqrt{2}-\sqrt{2}i.
$$
**Step 1. Express $$ z $$ in polar form.**

The modulus is
$$
r = \sqrt{(-\sqrt{2})^2+(-\sqrt{2})^2}=\sqrt{2+2}=\sqrt{4}=2.
$$
The argument $$ \theta $$ satisfies
$$
\tan\theta=\frac{-\sqrt{2}}{-\sqrt{2}}=1.
$$
Since both the real and imaginary parts are negative, $$ z $$ is in the third quadrant, so
$$
\theta=\pi+\frac{\pi}{4}=\frac{5\pi}{4}.
$$
Thus,
$$
z=2\left(\cos\frac{5\pi}{4}+i\sin\frac{5\pi}{4}\right).
$$

**Step 2. Apply de Moivre's theorem to compute $$ z^9 $$.**

Using de Moivre's theorem,
$$
z^9=2^9\left(\cos\left(9\cdot\frac{5\pi}{4}\right)+i\sin\left(9\cdot\frac{5\pi}{4}\right)\right).
$$
Calculating the power of the modulus,
$$
2^9=512.
$$
Calculating the argument,
$$
9\cdot\frac{5\pi}{4}=\frac{45\pi}{4}.
$$
We reduce the angle modulo $$ 2\pi $$ (where $$ 2\pi = \frac{8\pi}{4} $$):
$$
\frac{45\pi}{4}=\frac{40\pi}{4}+\frac{5\pi}{4}=10\pi+\frac{5\pi}{4}.
$$
Since adding $$ 10\pi $$ (an even multiple of $$ \pi $$) does not change the sine and cosine values modulo $$ 2\pi $$, the effective angle is
$$
\frac{5\pi}{4}.
$$

**Step 3. Write the final result in rectangular form.**

Thus,
$$
z^9=512\left(\cos\frac{5\pi}{4}+i\sin\frac{5\pi}{4}\right).
$$
Recall that
$$
\cos\frac{5\pi}{4}=-\frac{1}{\sqrt{2}} \quad \text{and} \quad \sin\frac{5\pi}{4}=-\frac{1}{\sqrt{2}}.
$$
So,
$$
z^9=512\left(-\frac{1}{\sqrt{2}}-i\frac{1}{\sqrt{2}}\right)=-\frac{512}{\sqrt{2}}-\frac{512}{\sqrt{2}}i.
$$
Simplify by rationalizing the denominator:
$$
\frac{512}{\sqrt{2}}=512\cdot\frac{\sqrt{2}}{2}=256\sqrt{2}.
$$
Thus,
$$
z^9=-256\sqrt{2}-256\sqrt{2}\, i.
$$

**Answer:**
$$
-256\sqrt{2}-256\sqrt{2}\, i.
$$
The answer is $$ -256\sqrt{2}-256\sqrt{2}\, i $$.

Using de Moivre’s theorem to compute we find that the answer is

None of the others

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Using de Moivre’s theorem to compute we find that the answer is None of the others