W15 Stepwise

Due date: Thursday 4/23, 11:59pm

Complex algebra

01

01

Complex arithmetic

Write each of these expressions in the form $a+bi$.

(a) $(1+3i)(5-i)$ $$ (b) ${\displaystyle \frac{2+5i}{-3+7i}}$

Link to original

Solution

Solutions---0290-01

02

02

Complex solutions of quadratic equations

Find all solutions and write them in the form $z=a+bi$.

$$2z^2+z+1=0$$

Link to original

Solution

Solutions---0290-02

Complex exponential

03

01

Complex forms - exponential to Cartesian

Write each number in the form $a+bi$.

(a) $2e^{i\frac{\pi }{4}}$ $$ (b) $e^{\ln 4+i\frac{\pi }{2}}$

Link to original

Solution

Solutions---0300-01

04

02

Polar and exponential form

Write down Euler’s Formula.

Now write $-5+5i$:

(i) in polar form $$ (ii) in exponential form

Link to original

Solution

Solutions---0300-02

Complex roots

05

01

Complex roots using polar

Find the three cube ($3^{\text{rd}}$) roots of $27i$.

Write your answer in the form $a+bi$.

Link to original

Solution

Solutions---0310-01