Chapter 3

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?

Evaluate the algebraic expressions. If \(f(x)=x^{2}+x-4,\) evaluate \(f(2 i)\)

Evaluate the algebraic expressions. If \(f(x)=x^{3}-2,\) evaluate \(f(i)\)

Evaluate the algebraic expressions. If \(f(x)=x^{2}+3 x+5,\) evaluate \(f(2+i)\)

Evaluate the algebraic expressions. If \(f(x)=2 x^{2}+x-3,\) evaluate \(f(2-3 i)\)

Evaluate the algebraic expressions. If \(f(x)=\frac{x+1}{2-x},\) evaluate \(f(5 i)\)

Evaluate the algebraic expressions. If \(f(x)=\frac{1+2 x}{x+3},\) evaluate \(f(4 i)\)

Plot the complex numbers on the complex plane. \(1-2 i\)

Plot the complex numbers on the complex plane. \(-2+3 i\)

Plot the complex numbers on the complex plane. \(i\)

Plot the complex numbers on the complex plane. \(-3-4 i\)

Perform the indicated operation and express the result as a simplified complex number. \((3+2 i)+(5-3 i)\)

Perform the indicated operation and express the result as a simplified complex number. \((-2-4 i)+(1+6 i)\)

Perform the indicated operation and express the result as a simplified complex number. \((-5+3 i)-(6-i)\)

Perform the indicated operation and express the result as a simplified complex number. \((2-3 i)-(3+2 i)\)

Perform the indicated operation and express the result as a simplified complex number. \((-4+4 i)-(-6+9 i)\)

Perform the indicated operation and express the result as a simplified complex number. \((2+3 i)(4 i)\)

Perform the indicated operation and express the result as a simplified complex number. \((5-2 i)(3 i)\)

For the following exercises, perform the indicated operation and express the result as a simplified complex number \((6-2 i)(5)\)

Perform the indicated operation and express the result as a simplified complex number. \((-2+4 i)(8)\)

Perform the indicated operation and express the result as a simplified complex number. \((2+3 i)(4-i)\)

Perform the indicated operation and express the result as a simplified complex number. \((-1+2 i)(-2+3 i)\)

Perform the indicated operation and express the result as a simplified complex number. \((4-2 i)(4+2 i)\)

Perform the indicated operation and express the result as a simplified complex number. \((3+4 i)(3-4 i)\)

Perform the indicated operation and express the result as a simplified complex number. \(\frac{3+4 i}{2}\)

Perform the indicated operation and express the result as a simplified complex number. \(\frac{6-2 i}{3}\)

Perform the indicated operation and express the result as a simplified complex number. \(\frac{-5+3 i}{2 i}\)

Perform the indicated operation and express the result as a simplified complex number. \(\frac{6+4 i}{i}\)

Perform the indicated operation and express the result as a simplified complex number. \(\frac{2-3 i}{4+3 i}\)

Perform the indicated operation and express the result as a simplified complex number. \(\frac{3+4 i}{2-i}\)

Perform the indicated operation and express the result as a simplified complex number. \(\frac{2+3 i}{2-3 i}\)

Perform the indicated operation and express the result as a simplified complex number. \(\sqrt{-9}+3 \sqrt{-16}\)

Perform the indicated operation and express the result as a simplified complex number. \(-\sqrt{-4}-4 \sqrt{-25}\)

Perform the indicated operation and express the result as a simplified complex number. \(\frac{2+\sqrt{-12}}{2}\)

Perform the indicated operation and express the result as a simplified complex number. \(\frac{4+\sqrt{-20}}{2}\)

Perform the indicated operation and express the result as a simplified complex number. \(i^{8}\)

Perform the indicated operation and express the result as a simplified complex number. \(i^{15}\)

Perform the indicated operation and express the result as a simplified complex number. \(i^{22}\)

Use a calculator to help answer the questions. Evaluate \((1+i)^{k}\) for \(k=4,8,\) and \(12 .\) Predict the value if \(k=16\)

Use a calculator to help answer the questions. Evaluate \((1-i)^{k}\) for \(k=2,6,\) and \(10 .\) Predict the value if \(k=14\)

Use a calculator to help answer the questions. Evaluate \((1+i)^{k}-(1-i)^{k}\) for \(k=4,8,\) and \(12 .\) Predict the value for \(k=16\)

Use a calculator to help answer the questions. Show that a solution of \(x^{6}+1=0\) is \(\frac{\sqrt{3}}{2}+\frac{1}{2} i\)

Use a calculator to help answer the questions. Show that a solution of \(x^{8}-1=0\) is \(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i\)

Evaluate the expressions, writing the result as a simplified complex number. \(\frac{1}{i}+\frac{4}{i^{3}}\)

Evaluate the expressions, writing the result as a simplified complex number. \(\frac{1}{i^{11}}-\frac{1}{i^{21}}\)

Evaluate the expressions, writing the result as a simplified complex number. \(i^{7}\left(1+i^{2}\right)\)

Evaluate the expressions, writing the result as a simplified complex number. \(i^{-3}+5 i^{7}\)