Notes: `(2 + 2i)^6` Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form.

$\displaystyle{z}={{\left({2}+{2}{i}\right)}}^{{6}}$

$\displaystyle{r}=\sqrt{{{{\left({2}\right)}}^{{2}}+{{\left({2}\right)}}^{{2}}}}=\sqrt{{8}}={2}\sqrt{{2}}$

$\displaystyle\theta={\arctan{{\left(\frac{{2}}{{2}}\right)}}}={\arctan{{\left({1}\right)}}}=\frac{\pi}{{4}}$

DeMoivre's Theorem

$\displaystyle{{z}}^{{n}}={{\left[{r}{\left({\cos{\theta}}+{i}{\sin{\theta}}\right)}\right]}}^{{n}}={{r}}^{{n}}{\left({\cos{{n}}}\theta+{i}{\sin{{n}}}\theta\right)}$

$\displaystyle{{z}}^{{6}}={\left[{2}\sqrt{{2}}{{\left({\cos{{\left(\frac{\pi}{{4}}\right)}}}+{i}{\sin{{\left(\frac{\pi}{{4}}\right)}}}\right]}}^{{6}}={{\left({2}\sqrt{{2}}\right)}}^{{6}}{\left[{\cos{{6}}}{\left(\frac{\pi}{{4}}\right)}+{i}{\sin{{6}}}{\left(\frac{\pi}{{4}}\right)}\right]}\right.}$

$\displaystyle{{z}}^{{6}}={512}{\left[{\cos{{\left(\frac{{{3}\pi}}{{2}}\right)}}}+{i}{\sin{{\left(\frac{{{3}\pi}}{{2}}\right)}}}\right]}$

$\displaystyle{{z}}^{{6}}={512}{\left[{0}+{\left(-{1}\right)}{i}\right]}$

$\displaystyle{{z}}^{{6}}=-{512}{i}$

$\displaystyle{z}={{\left({2}+{2}{i}\right)}}^{{6}}$

$\displaystyle{r}=\sqrt{{{{\left({2}\right)}}^{{2}}+{{\left({2}\right)}}^{{2}}}}=\sqrt{{8}}={2}\sqrt{{2}}$

$\displaystyle\theta={\arctan{{\left(\frac{{2}}{{2}}\right)}}}={\arctan{{\left({1}\right)}}}=\frac{\pi}{{4}}$

DeMoivre's Theorem

$\displaystyle{{z}}^{{n}}={{\left[{r}{\left({\cos{\theta}}+{i}{\sin{\theta}}\right)}\right]}}^{{n}}={{r}}^{{n}}{\left({\cos{{n}}}\theta+{i}{\sin{{n}}}\theta\right)}$

$\displaystyle{{z}}^{{6}}={512}{\left[{\cos{{\left(\frac{{{3}\pi}}{{2}}\right)}}}+{i}{\sin{{\left(\frac{{{3}\pi}}{{2}}\right)}}}\right]}$

$\displaystyle{{z}}^{{6}}={512}{\left[{0}+{\left(-{1}\right)}{i}\right]}$

$\displaystyle{{z}}^{{6}}=-{512}{i}$

Notes

Friday, 3 November 2017

$\displaystyle{{\left({2}+{2}{i}\right)}}^{{6}}$ Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form.

No comments:

Post a Comment

Is there any personification in "The Tell-Tale Heart"?

Friday, 3 November 2017

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form.

No comments:

Post a Comment

Is there any personification in &quot;The Tell-Tale Heart&quot;?

$\displaystyle{{\left({2}+{2}{i}\right)}}^{{6}}$ Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form.

Is there any personification in "The Tell-Tale Heart"?