### About Complex Numbers:

When we want to take the square root of a negative number, we previously stopped and wrote “no real solution”. Once we introduce the complex number system and the imaginary unit “i”, it becomes possible to give a solution. We define i, the imaginary unit to be the square root of negative one. We can use this to take the square root of any negative number.

Test Objectives

- Demonstrate the ability to take the square root of a negative number
- Demonstrate the ability to perform operations with complex numbers
- Demonstrate the ability to simplify powers of i

#1:

Instructions: Write each as the product of i and a real number.

a) $$\sqrt{-45}$$

b) $$\sqrt{-125}$$

c) $$\sqrt{-147}$$

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#2:

Instructions: Simplify.

a) $$(-1 + 4i) - (2 + 3i) - (3 + i)$$

b) $$(-7 - 6i) + (-3 - 3i) - (-3 - 5i)$$

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#3:

Instructions: Simplify.

a) $$-3(8 + 5i)(2 + 3i)$$

b) $$(-7 + 7i)(6 + 4i)(1 - i)$$

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#4:

Instructions: Simplify.

a) $$\frac{5 + 3i}{1 + 2i}$$

b) $$\frac{10 + 5i}{-7 + 4i}$$

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#5:

Instructions: Simplify.

a) $$i^{103}$$

b) $$i^{58}$$

c) $$i^{260}$$

d) $$i^{72}$$

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Written Solutions:

#1:

Solutions:

a) $$3i\sqrt{5}$$

b) $$5i\sqrt{5}$$

c) $$7i\sqrt{3}$$

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#2:

Solutions:

a) $$-6$$

b) $$-7 - 4i$$

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#3:

Solutions:

a) $$-3 - 102i$$

b) $$-56 + 84i$$

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#4:

Solutions:

a) $$\frac{11}{5} - \frac{7i}{5}$$

b) $$\frac{-10}{13} - \frac{15i}{13}$$

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#5:

Solutions:

a) $$-i$$

b) $$-1$$

c) $$1$$

d) $$1$$