Notes: `(3 - 2i)^5` Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form.

$\displaystyle{z}={{\left({3}-{2}{i}\right)}}^{{5}}$

$\displaystyle{r}=\sqrt{{{{\left({3}\right)}}^{{2}}+{{\left(-{2}\right)}}^{{2}}}}=\sqrt{{{9}+{4}}}=\sqrt{{13}}$

$\displaystyle\theta={\arctan{{\left(-\frac{{2}}{{3}}\right)}}}=-{.5880}$

DeMoivre's Theorem

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

$\displaystyle{{z}}^{{5}}={{\left[\sqrt{{13}}{\left({\cos{{\left(-{.5880}\right)}}}+{i}{\sin{{\left(-{.5880}\right)}}}\right)}\right]}}^{{5}}$

$\displaystyle{{z}}^{{5}}={{\left(\sqrt{{13}}\right)}}^{{5}}{\left[{\cos{{5}}}{\left(-{.5880}\right)}+{i}{\sin{{5}}}{\left(-{.5880}\right)}\right]}$

$\displaystyle{{z}}^{{5}}={169}\sqrt{{13}}{\left[{\cos{{\left(-{2.94}\right)}}}+{i}{\sin{{\left(-{2.94}\right)}}}\right]}$

$\displaystyle{{z}}^{{5}}=-{597.00}-{122.00}{i}$

$\displaystyle{z}={{\left({3}-{2}{i}\right)}}^{{5}}$

$\displaystyle{r}=\sqrt{{{{\left({3}\right)}}^{{2}}+{{\left(-{2}\right)}}^{{2}}}}=\sqrt{{{9}+{4}}}=\sqrt{{13}}$

$\displaystyle\theta={\arctan{{\left(-\frac{{2}}{{3}}\right)}}}=-{.5880}$

DeMoivre's Theorem

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

$\displaystyle{{z}}^{{5}}={{\left[\sqrt{{13}}{\left({\cos{{\left(-{.5880}\right)}}}+{i}{\sin{{\left(-{.5880}\right)}}}\right)}\right]}}^{{5}}$

$\displaystyle{{z}}^{{5}}={{\left(\sqrt{{13}}\right)}}^{{5}}{\left[{\cos{{5}}}{\left(-{.5880}\right)}+{i}{\sin{{5}}}{\left(-{.5880}\right)}\right]}$

$\displaystyle{{z}}^{{5}}={169}\sqrt{{13}}{\left[{\cos{{\left(-{2.94}\right)}}}+{i}{\sin{{\left(-{2.94}\right)}}}\right]}$

$\displaystyle{{z}}^{{5}}=-{597.00}-{122.00}{i}$

Notes

Wednesday, 3 August 2016

$\displaystyle{{\left({3}-{2}{i}\right)}}^{{5}}$ Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form.

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

Wednesday, 3 August 2016

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

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Post a Comment

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

$\displaystyle{{\left({3}-{2}{i}\right)}}^{{5}}$ 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"?