In Exercises compnumbasicfirst - compnumbasiclast, use the given complex numbers \(z\) and \(w\) to find and simplify the following.
  • \(z+w\)
  • \(zw\)
  • \(z^2\)
  • \(\frac {1}{z}\)
  • \(\frac {z}{w}\)
  • \(\frac {w}{z}\)
  • \(\overline {z}\)
  • \(z\overline {z}\)
  • \((\overline {z})^2\)
\(z = 2+3i\), \(w = 4i\)

  • \(z+w = 2+7i\)
  • \(zw = -12+8i\)
  • \(z^2 = -5 + 12i\)
  • \(\frac {1}{z} = \frac {2}{13} - \frac {3}{13} \, i\)
  • \(\frac {z}{w} = \frac {3}{4} - \frac {1}{2} \, i\)
  • \(\frac {w}{z} = \frac {12}{13} + \frac {8}{13} \,i\)
  • \(\overline {z} = 2-3i\)
  • \(z\overline {z} = 13\)
  • \((\overline {z})^2 = -5-12i\)

\(z = 1+i\), \(w = -i\)

  • \(z+w = \answer {1}\)
  • \(zw = \answer {1-i}\)
  • \(z^2 = \answer {2i}\)
  • \(\frac {1}{z} = \answer {\frac {1}{2} - \frac {1}{2}i}\)
  • \(\frac {z}{w} = \answer {-1+i}\)
  • \(\frac {w}{z} = \answer {-\frac {1}{2} - \frac {1}{2}i}\)
  • \(\overline {z} = \answer {1-i}\)
  • \(z\overline {z} = \answer {2}\)
  • \((\overline {z})^2 = \answer {-2i}\)
\(z = i\), \(w = -1+2i\)

  • \(z+w = -1+3i\)
  • \(zw = -2-i\)
  • \(z^2 = -1\)
  • \(\frac {1}{z} = -i\)
  • \(\frac {z}{w} = \frac {2}{5} - \frac {1}{5} \, i\)
  • \(\frac {w}{z} = 2+i\)
  • \(\overline {z} = -i\)
  • \(z\overline {z} = 1\)
  • \((\overline {z})^2 = -1\)
\(z = 4i\), \(w = 2-2i\)
  • \(z+w = \answer {2+2i}\)
  • \(zw = \answer {8+8i}\)
  • \(z^2 = \answer {-16}\)
  • \(\frac {1}{z} = \answer {-\frac {1}{4}i}\)
  • \(\frac {z}{w} = \answer {-1+i}\)
  • \(\frac {w}{z} = \answer {-\frac {1}{2} - \frac {1}{2}i}\)
  • \(\overline {z} = \answer {-4i}\)
  • \(z\overline {z} = \answer {16}\)
  • \((\overline {z})^2 = \answer {-16}\)
\(z = 3-5i\), \(w = 2+7i\)

  • \(z+w = 5+2i\)
  • \(zw = 41+11i\)
  • \(z^2 = -16-30i\)
  • \(\frac {1}{z} = \frac {3}{34} + \frac {5}{34} \,i\)
  • \(\frac {z}{w} = -\frac {29}{53} - \frac {31}{53} \, i\)
  • \(\frac {w}{z} = -\frac {29}{34} + \frac {31}{34} \,i\)
  • \(\overline {z} = 3+5i\)
  • \(z\overline {z} = 34\)
  • \((\overline {z})^2 = -16+30i\)
\(z = -5+i\), \(w = 4+2i\)
  • \(z+w = \answer {-1+3i}\)
  • \(zw = \answer {-22-6i}\)
  • \(z^2 = \answer {24-10i}\)
  • \(\frac {1}{z} = \answer {-\frac {5}{26} - \frac {1}{26}i}\)
  • \(\frac {z}{w} = \answer {-\frac {9}{10} + \frac {7}{10}i}\)
  • \(\frac {w}{z} = \answer {-\frac {9}{13} - \frac {7}{13}i}\)
  • \(\overline {z} = \answer {-5-i}\)
  • \(z\overline {z} = \answer {26}\)
  • \((\overline {z})^2 = \answer {24+10i}\)
\(z = \sqrt {2} - i\sqrt {2}\), \(w = \sqrt {2} + i\sqrt {2}\)

  • \(z+w = 2\sqrt {2}\)
  • \(zw = 4\)
  • \(z^2 = -4i\)
  • \(\frac {1}{z} = \frac {\sqrt {2}}{4} + \frac {\sqrt {2}}{4} \,i\)
  • \(\frac {z}{w} = -i\)
  • \(\frac {w}{z} = i\)
  • \(\overline {z} = \sqrt {2}+i\sqrt {2}\)
  • \(z\overline {z} = 4\)
  • \((\overline {z})^2 = 4i\)
\(z = 1 - i\sqrt {3}\), \(w = -1 - i\sqrt {3}\)
  • \(z+w = \answer {-2i\sqrt {3}}\)
  • \(zw = \answer {-4}\)
  • \(z^2 = \answer {-2-2i\sqrt {3}}\)
  • \(\frac {1}{z} = \answer {\frac {1}{4} + \frac {\sqrt {3}}{4}i}\)
  • \(\frac {z}{w} = \answer {\frac {1}{2} + \frac {\sqrt {3}}{2}i}\)
  • \(\frac {w}{z} = \answer {\frac {1}{2} - \frac {\sqrt {3}}{2}i}\)
  • \(\overline {z} = \answer {1+i\sqrt {3}}\)
  • \(z\overline {z} = \answer {4}\)
  • \((\overline {z})^2 = \answer {-2+2i\sqrt {3}}\)
\(z = \frac {1}{2} + \frac {\sqrt {3}}{2} \, i\), \(w = -\frac {1}{2} + \frac {\sqrt {3}}{2} \,i\)

  • \(z+w = i\sqrt {3}\)
  • \(zw = -1\)
  • \(z^2 = -\frac {1}{2} + \frac {\sqrt {3}}{2} \,i\)
  • \(\frac {1}{z} = \frac {1}{2} - \frac {\sqrt {3}}{2} \, i\)
  • \(\frac {z}{w} = \frac {1}{2} - \frac {\sqrt {3}}{2} \, i\)
  • \(\frac {w}{z} = \frac {1}{2} + \frac {\sqrt {3}}{2} \, i\)
  • \(\overline {z} = \frac {1}{2} - \frac {\sqrt {3}}{2} \, i\)
  • \(z\overline {z} = 1\)
  • \((\overline {z})^2 = -\frac {1}{2} - \frac {\sqrt {3}}{2} \, i\)
\(z = -\frac {\sqrt {2}}{2} + \frac {\sqrt {2}}{2} \, i\), \(w = -\frac {\sqrt {2}}{2} - \frac {\sqrt {2}}{2} \, i\)
  • \(z + w = \answer {-\sqrt {2}}\)
  • \(zw = \answer {1}\)
  • \(z^2 = \answer {-i}\)
  • \(\frac {1}{z} = \answer {-\frac {\sqrt {2}}{2} - \frac {\sqrt {2}}{2}i}\)
  • \(\frac {z}{w} = \answer {-i}\)
  • \(\frac {w}{z} = \answer {i}\)
  • \(\overline {z} = \answer {-\frac {\sqrt {2}}{2} - \frac {\sqrt {2}}{2}i}\)
  • \(z\overline {z} = \answer {1}\)
  • \((\overline {z})^2 = \answer {i}\)
In Exercises rootsofnegfirst - rootsofneglast, simplify the quantity.
\(\sqrt {-49}\)

\(7i\)
\(\sqrt {-9} = \answer {3i}\)
\(\sqrt {-25}\sqrt {-4}\)

\(10i^2 = -10\)
\(\sqrt {(-25)(-4)} = \answer {10}\)
\(\sqrt {-9}\sqrt {-16} = \answer {-12}\)
\(\sqrt {(-9)(-16)}\)

\(\sqrt {144} = 12\)
\(\sqrt {-(-9)}\)

\(\sqrt {9} = 3\)
\(-\sqrt {(-9)} = \answer {-3i}\)

We know that \(i^{2} = -1\) which means \(i^{3} = i^{2} \cdot i = (-1) \cdot i = -i\) and \(i^{4} = i^{2} \cdot i^{2} = (-1)(-1) = 1\). In Exercises powerofifirst - powerofilast, use this information to simplify the given power of \(i\).

\(i^{5}\)

\(i^{5} = i^{4} \cdot i = 1 \cdot i = i\)
\(i ^{6}\)

\(i ^{6} = i^{4} \cdot i^{2} = 1 \cdot (-1) = -1\)
\(i^{7} = \answer {-i}\)
\(i^{8} = \answer {1}\)
\(i^{15}\)

\(i^{15} = \left (i^{4}\right )^{3} \cdot i^{3} = 1 \cdot (-i) = -i\)
\(i^{26} = \answer {-1}\)
\(i^{117}\)

\(i^{117} = \left (i^{4}\right )^{29} \cdot i = 1 \cdot i = i\)
\(i^{304} = \answer {1}\)

In Exercises solvecomplexfirst - solvecomplexlast, find all complex solutions.

\(3x^2 + 6 = 4x\)

\(x = \frac {2 \pm i\sqrt {14}}{3}\)
\(15t^2+2t+5= 3t(t^2+1)\)

\(t = 5, \pm \frac {i \sqrt {3}}{3}\)
\(3y^2 + 4 = y^4\)

\(y = \pm 2, \pm i\)
\(\frac {2}{1-w}= w\)

\(w = \frac {1 \pm i \sqrt {7}}{2}\)
\(\frac {y}{3} - \frac {3}{y} = y\)

\(y = \pm \frac {3i \sqrt {2}}{2}\)
\(\frac {x^3}{2x-1} = \frac {x}{3}\)

\(x= 0, \frac {1 \pm i\sqrt {2}}{3}\)
\(x =\frac {2}{\sqrt {5} - x}\)

\(x = \frac {\sqrt {5} \pm i\sqrt {3}}{2}\)
\(\frac {5y^4 + 1}{y^2-1} = 3y^2\)

\(y = \pm i, \pm \frac {i\sqrt {2}}{2}\)
\(z^{4} = 16\)

\(z = \pm 2, \pm 2i\)
Multiply and simplify: \(\left (x - [3 - i\sqrt {23}]\right )\left (x - [3+i\sqrt {23}]\right )\)

\(x^2 - 6x + 32\)