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) $$\frac{11}{5}- \frac{7}{5}i$$
b) $$-\frac{10}{13}- \frac{15}{13}i$$
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#5:
Solutions:
a) $$-i$$
b) $$-1$$
c) $$1$$
d) $$1$$
