Notes
Practice Problems
\(\textbf{1)}\) Find the absolute value of \(3-4i\)
The answer is \(5\)
\(\textbf{2)}\) Find the absolute value of \(\sqrt{7}+i\sqrt{7}\)
The answer is \(\sqrt{14}\)
\(\textbf{3)}\) Find the absolute value of \(5\left(\cos{\frac{25 \pi }{3}}+i \sin{\frac{25\pi}{3}}\right)\)
The answer is \(5\)
\(\textbf{4)}\) Find the absolute value of \(\sqrt{6}\left(\cos \pi + i \sin \pi\right)\)
The answer is \(\sqrt{6}\)
\(\textbf{5)}\) Convert \(2\sqrt{3}+2i\) to polar form
The answer is \(4\left(\cos{\frac{ \pi }{6}}+i \sin{\frac{\pi}{6}}\right)\)
\(\textbf{6)}\) Convert \(-4+4i\) to polar form
The answer is \(4\sqrt{2}\left(\cos{\frac{3 \pi }{4}}+i \sin{\frac{3\pi}{4}}\right)\)
\(\textbf{7)}\) Convert \(8\) to polar form
The answer is \(8\left(\cos 0 + i \sin 0\right)\)
\(\textbf{8)}\) Convert \(\displaystyle -\frac{3\sqrt{2}}{2}-\frac{3\sqrt{2}}{2}i\) to polar form
The answer is \(3\left(\cos{\frac{5 \pi }{4}}+i \sin{\frac{5\pi}{4}}\right)\)
\(\textbf{9)}\) Convert \(\sqrt{10}\left(\cos{\frac{ \pi }{2}}+i \sin{\frac{\pi}{2}}\right)\) to rectangular form
The answer is \(i\sqrt{10}\)
\(\textbf{10)}\) Convert \(2\left(\cos{\frac{ 4\pi }{3}}+i \sin{\frac{4\pi}{3}}\right)\) to rectangular form
The answer is \(-1-\sqrt{3}\)
\(\textbf{11)}\) Convert \(12\left(\cos{\pi}+i \sin{\pi}\right)\) to rectangular form
The answer is \(-12\)
\(\textbf{12)}\) Convert \(5\left(\cos{\frac{ 5\pi }{6}}+i \sin{\frac{5\pi}{6}}\right)\) to rectangular form
The answer is \(\displaystyle -\frac{5\sqrt{3}}{2}-\frac{5}{2}i\)
\(\textbf{13)}\) Simplify \(\left(6+5i\right)\left(2-3i\right)\)
The answer is \(27-8i\)
\(\textbf{14)}\) Simplify \(\left(3+2i\right)\left(5-4i\right)\)
The answer is \(23-2i\)
\(\textbf{15)}\) Simplify \(\left(4+i\right)\left(1-4i\right)\)
The answer is \(8-15i\)
\(\textbf{16)}\) Simplify \(\displaystyle \frac{3+1}{2-i}\)
The answer is \(1+i\)
\(\textbf{17)}\) Simplify \(\displaystyle \frac{7-i}{4+3i}\)
The answer is \(1-i\)
\(\textbf{18)}\) Simplify \(\displaystyle \frac{4+3i}{3-4i}\)
The answer is \(i \)
\(\textbf{19)}\) Simplify \(\left(1-i\right)^3\)
The answer is \(-2-2i\)
\(\textbf{20)}\) Simplify \(3\sqrt{12}\left(\cos{\frac{ 3\pi }{4}}+i \sin{\frac{3\pi}{4}}\right)\cdot 8\left(\cos{\frac{ \pi }{2}}+i \sin{\frac{\pi}{2}}\right)\)
The answer is \(24\sqrt2\left(\cos{\frac{ 5\pi }{4}}+i \sin{\frac{5\pi}{4}}\right)\)
\(\textbf{21)}\) Simplify \(5\left(\cos{\frac{ \pi }{4}}+i \sin{\frac{\pi}{4}}\right)\cdot 7\sqrt{3}\left(\cos{\frac{ 5\pi }{6}}+i \sin{\frac{5\pi}{6}}\right)\)
The answer is \(7\sqrt3\left(\cos{\frac{ 5\pi }{6}}+i \sin{\frac{5\pi}{6}}\right)\)
\(\textbf{22)}\) Simplify \(\displaystyle\frac{6\left(\cos{\frac{ \pi }{6}}+i \sin{\frac{\pi}{6}}\right)}{ \sqrt{3}\left(\cos{\frac{ 5\pi }{6}}+i \sin{\frac{5\pi}{6}}\right)}\)
The answer is \(2\sqrt3\left(\cos-{\frac{ 2\pi }{3}}+i \sin-{\frac{2\pi}{3}}\right)\)
\(\textbf{23)}\) Simplify \(\displaystyle\frac{10\left(\cos{\frac{ \pi }{8}}+i \sin{\frac{\pi}{8}}\right)}{ 5\left(\cos{\frac{ \pi }{10}}+i \sin{\frac{\pi}{10}}\right)}\)
The answer is \(2\left(\cos-{\frac{ \pi }{40}}+i \sin-{\frac{\pi}{40}}\right)\)
\(\textbf{24)}\) Simplify \(\left(3\left(\cos{\frac{ 5\pi }{6}}+i \sin{\frac{5\pi}{6}}\right)\right)^3\)
The answer is \(27\left(\cos{\frac{ 5\pi }{2}}+i \sin{\frac{5\pi}{2}}\right)\)
\(\textbf{25)}\) Simplify \(\left(2\left(\cos{\frac{ 5\pi }{7}}+i \sin{\frac{5\pi}{7}}\right)\right)^3\)
The answer is \(8\left(\cos{\frac{ 15\pi }{7}}+i \sin{\frac{15\pi}{7}}\right)\)
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