\( 5 \angle+81.87^{\circ}\left(4-j 3+\frac{3 \sqrt{2} \angle-45^{\circ}}{7-j 1}\right) \)
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Find the angle between $\mathrm{v}$ and $\mathrm{w}$. Round to the nearest tenth of a degree. $$ \mathbf{v}=3 \mathbf{j}, \quad \mathbf{w}=4 \mathbf{i}+5 \mathbf{j} $$
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Express the roots of $(-14+j 3)^{\frac{-2}{5}}$ in polar form. $$ \begin{aligned} (-14+j 3)=& \sqrt{205} \angle 167.905^{\circ} \\ (-14+j 3)^{\frac{-2}{5}}=\sqrt{205^{\frac{-2}{5}}} \angle\left[\left(-\frac{2}{5}\right) \times 167.905^{\circ}\right] \\ =& 0.3449 \angle-67.164^{\circ} \\ & \text { or } 0.3449 \angle-67^{\circ} 10^{\prime} \end{aligned} $$ There are five roots to this complex number, $$ \left(x^{\frac{-2}{5}}=\frac{1}{x^{\frac{2}{5}}}=\frac{1}{\sqrt[5]{x^{2}}}\right) $$ The roots are symmetrically displaced from one another $(360 / 5)^{\circ}$, i.e. $72^{\circ}$ apart round an Argand diagram. Thus the required roots are $0.3449 \angle-67^{\circ} 10^{\prime}$, $0.3449 \angle 4^{\circ} 50^{\prime}, \quad 0.3449 \angle 76^{\circ} 50^{\prime}, \quad 0.3449 \angle 148^{\circ} 50^{\prime}$ and $0.3449 \angle 220^{\circ} 50^{\prime}$
Determine the value of $(-7+j 5)^{4}$ expressing the result in polar and rectangular forms. $$ \begin{aligned} (-7+j 5) &=\sqrt{\left[(-7)^{2}+5^{2}\right]} \angle \tan ^{-1} \frac{5}{-7} \\ &=\sqrt{74} \angle 144.46^{\circ} \end{aligned} $$ (Note, by considering the Argand diagram, $-7+j 5$ must represent an angle in the second quadrant and not in the fourth quadrant.) Applying de Moivre's theorem: $$ \begin{aligned} (-7+j 5)^{4} &=\left[\sqrt{74} \angle 144.46^{\circ}\right]^{4} \\ &=\sqrt{74^{4}} \angle 4 \times 144.46^{\circ} \\ &=5476 \angle 577.84^{\circ} \\ &=5476 \angle 217.84^{\circ} \end{aligned} $$ or $5476 \angle 217^{\circ} 50^{\prime}$ in polar form Since $r \angle \theta=r \cos \theta+j r \sin \theta$ $$ \begin{aligned} 5476 \angle 217.84^{\circ}=5476 \cos 217.84^{\circ} \\ &+j 5476 \sin 217.84^{\circ} \\=&-4325-j 3359 \end{aligned} $$ e. $\quad(-7+j 5)^{4}=-4325-j 3359$ in rectangular form
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