Explore

The Complex Number System Solutions

• The complex number system consists of numbers in the form $\underline{\mathbf{ }\mathbf{ }\mathbf{ }\mathit{a}\mathbf{+}\mathit{b}\mathit{i}\mathbf{ }\mathbf{ }}$, where a and b are part of the     real    number system, or $a\in \mathrm{ℝ}, b\in \mathrm{ℝ}$.
•    Complex numbers ℂ    includes the set of real numbers ℝ, and the set of imaginary numbers i.
• The number i can only be combined with    like terms   , meaning other imaginary numbers.
• This means that a real number and an imaginary number can only be written as an    expression   .
  •    Real numbers    are written as $a+bi$ where $b=0$ (or a).
  •    Pure imaginary numbers    numbers are written as $a+bi$ where $a=0$ (or bi).

Example 6

Classify by all sets to which it belongs: real, pure imaginary, complex. Then add A–D to the Complex Number diagram on the previous page.

Checkpoint: The Complex Number System Solution

Classify by all sets to which it belongs: real, pure imaginary, complex. Then add to the Complex Number diagram (in the guided notes).

1. $26i$ 
   pure imaginary, complex

2. $3 + 7i$ 
   complex

Note

Q: Are real numbers complex numbers? Are imaginary numbers complex numbers? Explain.

A: Both real and imaginary numbers are also complex numbers. You can see this in the diagram. It shows the real numbers and imaginary numbers inside the complex number box. 

Back to Top

Practice 1 Solutions

Evaluate.

Note

Use the Formula Sheet to help with the rules for the imaginary unit i.

1. $i^{29}$

$$\begin{gathered} 29÷4=7 \mathrm{R}1 \\ {\left(i^4\right)}^7·i \\ 1·i \end{gathered}$$

i

2. $i^{75}$

$$\begin{gathered} 75÷4=18 \mathrm{R}3 \\ {\left(i^4\right)}^{18}·i^3 \\ 1·–i \end{gathered}$$

– i

3. $i^{16}$

$$\begin{gathered} 16÷4=4 \\ {\left(i^4\right)}^4 \end{gathered}$$

1

4. $i^{50}$

$$\begin{gathered} 50÷4=12 \mathrm{R}2 \\ {\left(i^4\right)}^{12}·i^2 \\ 1·–1 \end{gathered}$$

–1

5. $i^{120}$

$$\begin{gathered} 120÷4=30 \\ {\left(i^4\right)}^{30} \end{gathered}$$

1

6. $i^{101}$

$$\begin{gathered} 101÷4=25 \mathrm{R}1 \\ {\left(i^4\right)}^{25}·i \\ 1·i \end{gathered}$$

i

7. $i^{38}$

$$\begin{gathered} 38÷4=9 \mathrm{R}2 \\ {\left(i^4\right)}^9·i^2 \\ 1·–1 \end{gathered}$$

–1

8. $i^{67}$

$$\begin{gathered} 67÷4=16 \mathrm{R}3 \\ {\left(i^4\right)}^{16}·i^3 \\ 1·–i \end{gathered}$$

– i

Simplify.

Note

Q: Why must the imaginary unit i be simplified out when the radicand is negative?

A: Because the rules to simplify radicals are only true for real numbers.

9. $\sqrt{–18}$

$i\sqrt{3^2·2}$

$3i\sqrt{2}$

10. $\sqrt{–36}·\sqrt{–16}$

$$\begin{gathered} i\sqrt{6^2}·i\sqrt{4^2} \\ 6i·4i \\ 24i^2 \end{gathered}$$

–24

Note

Remember to simplify out $\sqrt{–1}$before multiplication of any radicands. If your answer is positive, you did not write $\sqrt{–1}$as i in your first step.

11. $\sqrt{–54}$

$i\sqrt{3^3·2}$

$3i\sqrt{6}$

12. $\sqrt{–32}$

$i\sqrt{2^5}$

$4i\sqrt{2}$

13. $\sqrt{–144}$

$i\sqrt{{12}^2}$

12i

14. $3\sqrt{–10}·7\sqrt{–35}$

$$\begin{gathered} 3i\sqrt{2·5}·7i\sqrt{5·7} \\ 21i^2\sqrt{5^2·2·7} \\ 105i^2\sqrt{14} \end{gathered}$$

$–105\sqrt{14}$

Note

Q: What is another way to write $i^2$?

A: Negative one.

15. $2\sqrt{–81}·\sqrt{–24}$

$$\begin{gathered} 2i\sqrt{9^2}·i\sqrt{2^3·3} \\ 18i·2i\sqrt{6} \\ 36i^2\sqrt{6} \end{gathered}$$

$–36\sqrt{6}$

16. $\sqrt{–98}$

$i\sqrt{7^2·2}$

$7i\sqrt{2}$

Simplify. Then classify by all sets to which it belongs: real, pure imaginary, complex.

Note

Q: What classification can you use for every number?

A: Complex

Q: Why is it a good math practice to simplify before you classify numbers?

A: Because the simplified answer may not be the exact classification you originally expected.

17. $\sqrt{12}$

$2\sqrt{3}$

Real, complex

18. $\sqrt{16}+\sqrt{–8}$

$4+2i\sqrt{2}$

Complex

19. $\sqrt{–121}$

11i

Pure imaginary, complex

20. $5+\sqrt{49}$

$$\begin{gathered} 5+7 \\ 12 \end{gathered}$$

Real, complex

21. $3i$

Pure imaginary, complex

22. $\sqrt{–25}$

5i

Pure imaginary, complex

23. $\sqrt{–49}·\sqrt{–4}$

$$\begin{gathered} 7i·2i \\ 14i^2 \\ 14\left(–1\right) \\ –14 \end{gathered}$$

Real, complex

24. $13–\sqrt{–9}$

$13–3i$

Complex

Back to Top

Mastery Check Solutions

Show What You Know

Complete the statement with always, sometimes, or never.
If your answer is sometimes or never, provide an example to back up your choice.

1. The imaginary unit i raised to a whole number power is    sometimes    a pure imaginary number.

Sample: $i^1= i$ ,  $i^2=–1$ ,  $i^3= –i$  , and $i^4=1$. The numbers –1 and 1 are real, not imaginary, while i and $–i$ are pure imaginary.

2. Complex numbers    always    include the sets of real and imaginary numbers.

3. The square root of a negative number is    never    a real number.

Sample:$\sqrt{–1}=i$, where i is an imaginary number.

4. Complex numbers are    sometimes    classified as real, complex numbers.

Sample: $a + bi$, when $a=0$ is pure imaginary. If an expression is in the form $a+bi$, this is complex only. If an expression is in the form $a+bi$, when $b=0$, the number is real, complex.

Note

These statements should help you think about how the imaginary numbers work within the complex number system. There are many examples you can share for A, C, and D. If you are not certain, use the Practice page to find specific examples.

Say What You Know

In your own words, talk about what you have learned using the objectives for this part of the lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

• • Classify complex numbers (real, imaginary, complex).
  • Define the imaginary unit as $i^2=–1$.
  • Simplify roots of negative numbers using the imaginary unit.
  • Use the pattern of i to simplify powers of the imaginary unit.

Back to Top

Practice 2 Solutions

Evaluate.

1. $i^2$

–1

2. $i^{78}$

$$\begin{gathered} 78÷4=19 \mathrm{R}2 \\ {\left(i^4\right)}^{19}·i^2 \\ 1·–1 \end{gathered}$$

–1

3. $i^{15}$

$$\begin{gathered} 15÷4=3 \mathrm{R}3 \\ {\left(i^4\right)}^3·i^3 \\ 1·–i \end{gathered}$$

– i

4. $i^{10}$

$$\begin{gathered} 10÷4=2 \mathrm{R}2 \\ {\left(i^4\right)}^2·i^2 \\ 1·–1 \end{gathered}$$

–1

5. $i^{61}$

$$\begin{gathered} 61÷4=15 \mathrm{R}1 \\ {\left(i^4\right)}^{15}·i \\ 1·i \end{gathered}$$

i

6. $i^{99}$

$$\begin{gathered} 99÷4=24 \mathrm{R}3 \\ {\left(i^4\right)}^{24}·i^3 \\ 1·–i \end{gathered}$$

– i

Simplify.

7. $3\sqrt{–5}·2\sqrt{–4}$

$$\begin{gathered} 3i\sqrt{5}·2i\sqrt{2^2} \\ 3i\sqrt{5}·4i \\ 12i^2\sqrt{5} \end{gathered}$$

$–12\sqrt{5}$

8. $6\sqrt{–40}$

$6i\sqrt{2^3·5}$

$12i\sqrt{10}$

9. $\sqrt{–12}·\sqrt{–3}$

$$\begin{gathered} i\sqrt{2^2·3}·i\sqrt{3} \\ {i}^2\sqrt{2^2·3^2} \\ 6i^2 \end{gathered}$$

–6

10. $\sqrt{–196}$

$i\sqrt{{14}^2}$

14i

11. $20\sqrt{–7}·\sqrt{–14}$

$$\begin{gathered} 20i\sqrt{7}·i\sqrt{2·7} \\ 20i^2\sqrt{7^2·2} \\ 140i^2\sqrt{2} \end{gathered}$$

$–140\sqrt{2}$

12. $\sqrt{–64}$

$i\sqrt{8^2}$

8i

Simplify. Then classify by all sets to which it belongs: real, pure imaginary, complex.

13. $\sqrt{–4}$

2i

Pure imaginary, complex

14. $\sqrt{–49}·\sqrt{–100}$

$$\begin{gathered} i\sqrt{49}·i\sqrt{100} \\ 7i·10i \\ 70i^2 \\ 70\left(–1\right) \\ –70 \end{gathered}$$

Real, complex

15. $–5\sqrt{50}·\sqrt{–8}$

$$\begin{gathered} \left(–5\right)5\sqrt{2}·2i\sqrt{2} \\ –25\sqrt{2}·2i\sqrt{2} \\ –50i\sqrt{2^2} \\ \left(–50i\right)\left(2\right) \\ –100i \end{gathered}$$

Pure imaginary, complex

16. $\sqrt{–9}·\sqrt{–81}$

$$\begin{gathered} 3i·9i \\ 27i^2 \\ –27 \end{gathered}$$

Real, complex

17. $\sqrt{121}–\sqrt{–20}$

$11–2i\sqrt{5}$

Complex

18. $\sqrt{–45}$

$3i\sqrt{5}$

Pure imaginary, complex

Back to Top

Targeted Review Solutions

Simplify.

1. $\sqrt{35x}·\sqrt{15x^3}$

$$\begin{gathered} \sqrt{5·7·3·5·x^4} \\ {3}^{\frac{1}{2}}·5^{\frac{2}{2}}·7^{\frac{1}{2}}·x^{\frac{4}{2}} \end{gathered}$$

$5x^2\sqrt{21}$

2. $\left(10+\sqrt{7}\right)–\left(8+12\sqrt{7}\right)$

$10+\sqrt{7}–8–12\sqrt{7}$ 

$2–11\sqrt{7}$

3. $\frac{2+\sqrt{5}}{10+\sqrt{6}}$

$$\begin{gathered} \frac{2+\sqrt{5}}{10+\sqrt{6}}\left(\frac{10–\sqrt{6}}{10–\sqrt{6}}\right) \\ \frac{20–2\sqrt{6}+10\sqrt{5}–\sqrt{30}}{100–6} \end{gathered}$$

$\frac{20–2\sqrt{6}+10\sqrt{5}–\sqrt{30}}{94}$

Solve.

4. $\sqrt[3]{x+3}=3$

$\begin{array}{rcl}{\left(\sqrt[3]{x+3}\right)}^3& =& {\left(3\right)}^3\\ x+3& =& 27\end{array}$

$x=24$

5. $\sqrt{x}+5\ge 7$

$\begin{array}{rcl}\sqrt{x}& \ge & 2\\ {\left(\sqrt{x}\right)}^2& \ge & {\left(2\right)}^2\end{array}$

$x\ge 4$

6. $\sqrt{2x–3}<7$

$\begin{array}{rcl}2x–3& \ge & 0\\ x& \ge & 1.5 \mathrm{principal} \mathrm{root}\\ {\left(\sqrt{2x–3}\right)}^2& <& {\left(7\right)}^2\\ 2x–3& <& 49\\ 2x& <& 52\\ x& <& 26\end{array}$

$\begin{array}{rcl}1.5& \le & x<26\end{array}$

Graph.

7. $y=\frac{4}{3}x–2$

$m=\frac{4}{3}, b=–2$

8. $y=–\frac{1}{x–3}–2$

$$\begin{gathered} \mathrm{asymptotes}: x=3, y=2 \\ a=–1 \end{gathered}$$

Multiple Choice

A

9. Simplify. $\sqrt[3]{48x^8{y}^3}$

1. $2x^{2}y\sqrt[3]{6x^2}$

2. $2x^{2}y\sqrt[3]{3x^2}$

3. $4x^4{y}^2\sqrt[3]{3y}$

4. $16x^{2}y\sqrt[3]{x^2}$

    

   $$\begin{gathered} 2^{\frac{4}{3}}{3}^{\frac{1}{3}}{x}^{\frac{8}{3}}{y}^{\frac{3}{3}} \\ {2}^{1\frac{1}{3}}{3}^{\frac{1}{3}}{x}^{2\frac{2}{3}}{y}^1 \\ 2x^{2}y\sqrt[3]{6x^2} \end{gathered}$$

Note

2. The coefficient of the radicand is missing a factor of 2.

3. This option uses an index of 2 rather than 3.

4. This option divides 48 by 3 rather than using the prime factorization and exponent rules.

C

10. Solve. $\sqrt{3x–1}–6=–2$

1. $\frac{1}{3}$

2. 5

3. $\frac{17}{3}$

4. no solution

    

   $\begin{array}{rcl}\sqrt{3x–1}& =& 4\\ {\left(\sqrt{3x–1}\right)}^2& =& {\left(4\right)}^2\\ 3x–1& =& 16\\ 3x& =& 17\\ x& =& \frac{17}{3}\end{array}$

Note

1. This option represents the restriction of the principal root.

2. This option occurs if 1 is subtracted rather than added to both sides.

4. The radical needs to be isolated on one side before determining if there is no solution.

C

11. Name the range using the graph of the rational function.

1. $\left\{y|y\in \mathrm{ℝ}, y\ne –1\right\}$

2. $\left\{y|y\in \mathrm{ℝ}, y\ne 0\right\}$

3. $\left\{y|y\in \mathrm{ℝ}, y\ne 4\right\}$

4. $\left\{y|y\in \mathrm{ℝ}, y\ne 5\right\}$

Note

1.  –1 represents the restriction on the domain.

2. This option is the x-axis, and not a restriction for the range.

4. This option is the value of the y-intercept.

B

12. Determine the equation that best represents the linear function.

1. $y=\frac{1}{4}x+2$

2. $y=–\frac{1}{4}x+2$

3. $y=–\frac{1}{5}x+2$

4. $y=\frac{1}{5}x+2$

Note

A, D) The slope of the graph is negative. 

C, D) Using $\left(–4, 3\right)$ and $\left(0, 2\right)$, the slope is $–\frac{1}{4}$.

Problem	1	2	3	4	5	6	7	8	9	10	11	12
Origin	L11	L12	L12	L13	L14	L14	A1	L10	L11	L13	L10	A1

L = Lesson in this level, A1 = Algebra 1: Principles of Secondary Mathematics, FD = Foundational Knowledge

Back to Top