Unit 1 - Number and Algebra
Required Prior Knowledge
Questions
Expand$$\left(2+3x\right)^{4}&&
Solutions
Get Ready
Questions
Investigate the powers of \(i\).
Find a general rule for \(i^{n}\) where \(n\in \mathbb{Z}\).
Use your rule to determine the value of \(i^{2012}\).
Solutions
Notes
In order to raise complex numbers to a power, we must treat them as a binomial.
In order to evaluate a square root of a complex number, we must equate real and imaginary parts.
Examples and Your Turns
Example
Find \(\left(2+3i\right)^{4}\) in the form \(a+bi\).
Your Turn
Find \(\left(1-2i\right)^{3}\) in the form \(a+bi\).
Your Turn
Find \(\left(3-2i\right)^{5}\) in the form \(a+bi\).
Your Turn
Given that \(z=1-2i\), verify that $$\left(z^{5}\right)^{*}=\left(z^{*}\right)^{5}$$
Example
Evaluate \(\sqrt{8-6i}\) in the form \(a+bi\).
Your Turn
Evaluate \(\sqrt{3-4i}\) in the form \(a+bi\).
Your Turn
Evaluate \(\sqrt{7-24i}\) in the form \(a+bi\).
Your Turn
(i) Find the square roots of \(16+30i\).
(ii) Use your answer to part (i) to solve the equation \(z^{2}-2z-\left(3+7.5i\right)=0\), giving your answers in the form \(a+bi\).
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.