Question 1

Which of the following statements correctly describes the set of rational numbers?

Accepted Answer

Rational numbers are numbers that can be written as a quotient of two integers, where the denominator is not zero.

Question 2

Which of the following is an example of an irrational number?

Accepted Answer

$$\pi$$

Question 3

Simplify $$\sqrt{-36}$$ and express your answer in standard form.

Accepted Answer

$$6i$$

Question 4

A medical researcher records a temperature change of $$-5$$ degrees Celsius. Which set(s) of numbers does $$-5$$ belong to?

Accepted Answer

Integers, rational numbers, and real numbers

Question 5

What is the result of $$(3 + 4i) + (5 - 2i)$$?

Accepted Answer

$$8 + 2i$$

Question 6

A chemist mixes two solutions, one represented by $$7 - 3i$$ and another by $$-2 + 5i. $$What is the sum of the two solutions?

Accepted Answer

$$5 + 2i$$

Question 7

Multiply the complex numbers $$(2 + 3i)$$ and $$(4 - 5i)$$ and express your answer in standard form.

Accepted Answer

$$8 + 12i - 10i - 15i^2$ = 8 + 2i + 15 = 23 + 2i$$

Question 8

Divide $$\frac{6}{2 + i}$$ and express your answer in standard form.

Accepted Answer

$$\frac{12 - 6i}{5}$$

Question 9

Simplify $$\sqrt{-50}$$ and express your answer in terms of $$i$$ and radicals.

Accepted Answer

$$5i\sqrt{2}$$

Question 10

A financial analyst calculates a net loss represented by $$-8. $$Which of the following sets does $$-8$$ NOT belong to?

Accepted Answer

Natural numbers

Question 11

What is the product of $$(5 + 2i)(5 - 2i)$$?

Accepted Answer

$$29$$

Question 12

Simplify $$\sqrt{-27} + \sqrt{-48}$$ and express your answer in terms of $$i$$ and radicals.

Accepted Answer

$$7i\sqrt{3}$$

Question 13

A physicist needs to calculate $$\sqrt{-64}$$ for a quantum mechanics problem. What is the value?

Accepted Answer

$$8i$$

Question 14

Which of the following is the complex conjugate of $$7 + 5i$$?

Accepted Answer

$$7 - 5i$$

Question 15

Simplify $$(-3 + \sqrt{-12})$$ and express your answer in standard form.

Accepted Answer

$$-3 + 2i\sqrt{3}$$

Question 16

A biologist records a measurement of $$\sqrt{49}. $$Which set(s) does this value belong to?

Accepted Answer

Natural numbers, whole numbers, integers, rational numbers, and real numbers

Question 17

What is the result of $$(-2 + \sqrt{-3})^2$?

Accepted Answer

$$1 - 4i\sqrt{3}$$

Question 18

Divide $$\frac{5i}{2 - i}$$ and express your answer in standard form.

Accepted Answer

$$1 + 2i$$

Question 19

Simplify $$\sqrt{-81} - \sqrt{-144}.$$

Accepted Answer

$$-3i$$

Question 20

A student calculates $$(-5 - \sqrt{-9})^2$. What is the result?

Accepted Answer

$$16 + 30i$$

1.4 Number Systems and Complex Numbers: Foundations and Operations

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