# Honors Algebra 2: Assignment 48 Unit Review

• Class: Honors Algebra 2
• Author: Peter Atlas
• Algebra and Trigonometry: Structure and Method, Brown

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Simplify Completely:

1. $$\displaystyle i ^{2039}$$
2. Solution

$$-i$$

3. $$\displaystyle ( 6 \mod 3) + (9 \mod 4)$$
4. Solution

1

5. $$\displaystyle 3\sqrt{-10} \cdot \sqrt{-15}$$
6. Solution

$$-15\sqrt{6}$$

7. $$\displaystyle \frac{\sqrt{-20}}{\sqrt{-5}}$$
8. Solution

2

9. $$\displaystyle 3\sqrt{-48} + 5\sqrt{-3}$$
10. Solution

$$17i\sqrt{3}$$

11. $$\displaystyle i ^{7} + i ^{8} + i ^{9}$$
12. Solution

1

13. $$\displaystyle \frac{1}{ i ^{21}}$$
14. Solution

$$-i$$

15. $$\displaystyle (7 - 3 i ) - (4 + i )$$
16. Solution

$$3 - 4i$$

17. If $$\displaystyle z = 4 + 7 i$$ and $$y = 2 - 3 i,$$ find $$z \cdot \bar{y}$$
18. Solution

$$\displaystyle -13 + 26i$$

19. $$\displaystyle \frac{1}{-3 + 2 i }$$
20. Solution

$$\displaystyle -\frac{3}{13} - \frac{2}{13}i$$

21. $$\displaystyle \frac{7}{5 - i \sqrt{3}}$$
22. Solution

$$\displaystyle \frac{5}{4} + \frac{\sqrt{3}}{4}i$$

23. $$\displaystyle \frac{8 - 3 i }{3 - 5 i }$$
24. Solution

$$\displaystyle \frac{39}{34} + \frac{31}{34}i$$

25. Show that the conjugate of the sum of two complex numbers is the sum of the conjugates of the numbers.
26. Solution

Let the first number be $$a + bi$$, and the second number be $$c + di$$. Then the conjugate of the sum is the conjugate of $$(a + c) + (b + d)i$$, which is $$(a + c) - (b + d)i$$. The sum of the conjugates of the numbers is $$a - bi + c - di = (a + c) - (b + d)i$$

27. Find real numbers $$x$$ and $$y$$ so that $$\displaystyle (x + y )i + 3x - 2 y = 7(4 + 3 i )$$
28. Solution

This becomes the system: \displaystyle \begin{align*} x + y &= 21 \\ 3x - 2y &= 28 \end{align*}. The solution is $$x = 14, y = 7.$$

29. $$\displaystyle \frac{2i}{\sqrt{-16}} + i \sqrt{-27}$$
30. Solution

$$\displaystyle \frac{1}{2} - 3\sqrt{3}$$

Factor Completely over the complex numbers:

1. $$\displaystyle 50x^{2} + 98$$
2. Solution

$$\displaystyle 2\left(5x + 7i\right)\left(5x - 7i \right)$$

3. $$\displaystyle v^{2} + 5$$
4. Solution

$$\displaystyle \left(v + i\sqrt{5}\right)\left(v - i\sqrt{5}\right)$$

5. $$\displaystyle x^{4} - y^{4}$$
6. Solution

$$\displaystyle (x + y)(x - y)(x + yi)(x - yi)$$

7. $$\displaystyle x^{4} + y^{4}$$
8. Solution

$$\displaystyle \left( \left( x - y\frac{\sqrt{2}}{2} \right) + y\frac{\sqrt{2}}{2}i \right) \left( \left( x + y\frac{\sqrt{2}}{2} \right) - y \frac{\sqrt{2}}{2}i \right) \left( \left( x + y \frac{\sqrt{2}}{2} \right) + y\frac{\sqrt{2}}{2}i \right) \left(\left(x - y\frac{\sqrt{2}}{2}\right) - y\frac{\sqrt{2}}{2}i \right)$$

20. Solve over the complex numbers

1. $$\displaystyle x^{2} = -121$$
2. Solution

$$\displaystyle x = \{ \pm 11i \}$$

3. $$\displaystyle x - 4 - \sqrt{x - 4} = 0$$
4. Solution

$$\displaystyle x = \{4, 5 \}$$

Given $$u = 1 + 3 i$$, on the same set of axes, graph

1. $$\displaystyle u$$
2. $$\displaystyle u^{2}$$
3. $$\displaystyle u \cdot \bar{u}$$
4. $$\displaystyle 2u - 4$$
Solutions to #21 - #24