Complex Numbers

1. Perform each operation and write the result in standard form \( a + bi\).  solution

(a)  \(\displaystyle{ (5 + 9i) + (6 - 8i) }\) 

(b)  \(\displaystyle{ (3 + 7i) - (2 - 4i) }\) 

(c)  \(\displaystyle{ -2i(3 - 6i) }\)

2. Perform each operation and write the result in standard form \( a + bi\).  solution

(a)  \(\displaystyle{ (5 + 2i) (2 - 3i) }\) 

(b)  \(\displaystyle{ (3 + 4i) (4 - 3i) }\) 

(c)  \(\displaystyle{ (\sqrt 3 - 7i)^2 }\)

3. Write the conjugate \(\bar{z}\) of each complex number \(z\). Then find \(z\bar{z}\).   solution

(a)  \( z=\sqrt{2}-5i\) 

(b) \( z= \frac{1}{2}+3i\)

4. Write each quotient in the standard form \( a + bi\).   solution

(a)  \(\displaystyle{ \frac {3}{-5i}}\) 

(b) \(\displaystyle{ \frac{5i}{2 - 3i} }\)  

(c)  \(\displaystyle{ \frac{3 - 5i}{4+i}}\)

5. Solve each equation.   solution

(a)  \( \displaystyle{ 4x^2 + 16 = 0 }\) 

(b) \(\displaystyle{ 8(x^2 - x) + 3 = x^2}\) 

6. Find each power of i and simplify.  solution

(a) \(\displaystyle{i^{16} + 2} \)   

(b) \(\displaystyle{i^{55}} \)  

(c)  \(\displaystyle{i^{113}+7} \)  

(d) \(\displaystyle{ i^{22} - 3i^6}\)