3.1 Complex numbers  (Page 4/8)

Explain how to add complex numbers.

What is the basic principle in multiplication of complex numbers?

Give an example to show the product of two imaginary numbers is not always imaginary.

What is a characteristic of the plot of a real number in the complex plane?

$\text{If }f(x)=x^2+x-4,$ evaluate $f(2i).$

$\text{If }f(x)=x^3-2,$ evaluate $f(i).$

$\text{If }f(x)=x^2+3x+5,$ evaluate $f(2+i).$

$\text{If }f(x)=2x^2+x-3,$ evaluate $f(2-3i).$

$\text{If }f(x)=\frac{x+1}{2-x},$ evaluate $f(5i).$

$\text{If }f(x)=\frac{1+2x}{x+3},$ evaluate $f(4i).$

$1-2i$

$-2+3i$

$i$

$-3-4i$

$\left(3+2i\right)+(5-3i)$

$\left(-2-4i\right)+\left(1+6i\right)$

$\left(-5+3i\right)-(6-i)$

$\left(2-3i\right)-(3+2i)$

$(-4+4i)-(-6+9i)$

$\left(2+3i\right)(4i)$

$\left(5-2i\right)(3i)$

$\left(6-2i\right)(5)$

$\left(-2+4i\right)\left(8\right)$

$\left(2+3i\right)(4-i)$

$\left(-1+2i\right)(-2+3i)$

$\left(4-2i\right)(4+2i)$

$\left(3+4i\right)\left(3-4i\right)$

$\frac{3+4i}{2}$

$\frac{6-2i}{3}$

$\frac{-5+3i}{2i}$

$\frac{6+4i}{i}$

$\frac{2-3i}{4+3i}$

$\frac{3+4i}{2-i}$

$\frac{2+3i}{2-3i}$

$\sqrt{-9}+3\sqrt{-16}$

$-\sqrt{-4}-4\sqrt{-25}$

$\frac{2+\sqrt{-12}}{2}$

$\frac{4+\sqrt{-20}}{2}$

$i^8$

$i^{15}$

$i^{22}$

Evaluate ${(1+i)}^k$ for $k=\text{4, 8, and 12}\text{.}$ Predict the value if $k=16.$

Evaluate ${(1-i)}^k$ for $k=\text{2, 6, and 10}\text{.}$ Predict the value if $k=14.$

Evaluate $(1+i{)}^k-(1-i{)}^k$ for $k=\text{4, 8, and 12}$ . Predict the value for $k=16.$

Show that a solution of $x^6+1=0$ is $\frac{\sqrt{3}}{2}+\frac{1}{2}i.$

Show that a solution of $x^8-1=0$ is $\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i.$

$\frac{1}{i}+\frac{4}{i^3}$

$\frac{1}{i^{11}}-\frac{1}{i^{21}}$

$i^7\left(1+i^2\right)$

$i^{\mathrm{-3}}+5i^7$

$\frac{\left(2+i\right)\left(4-2i\right)}{(1+i)}$

$\frac{\left(1+3i\right)\left(2-4i\right)}{(1+2i)}$

$\frac{{\left(3+i\right)}^2}{{\left(1+2i\right)}^2}$

$\frac{3+2i}{2+i}+\left(4+3i\right)$

$\frac{4+i}{i}+\frac{3-4i}{1-i}$

$\frac{3+2i}{1+2i}-\frac{2-3i}{3+i}$