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# Algebra 2/Trig Name: Unit 3 Notes Packet Date: Period:

Powers, Roots and Radicals

(1) Homework Packet

(2) Homework Packet

(3) Homework Packet

(4) Page 277 #4 – 10

(5) Page 277 – 278 #37 – 61 Odd

(6) Page 277 – 278 #38 – 60 Even

(7) Worksheet evens

(8) Worksheet odds

(9) Page 411 #5 – 20

(10) Page 411 – 412 #22 – 79 Column

(11) Page 411 – 412 #23 – 47 Column, #51, 54, #57 – 73 Column, #77, 80

(12) Page 411 – 412 #25 – 49 Column, #52, 55, #59 – 81 Column

(13) Chapter Review ***TEST TOMORROW***

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5.3 and Supplement Simplifying Radicals (Add, Sub, Conjugates – no variables) (R,I,E/3)

Perfect Squares: A number whose square roots are integers or quotients of integers

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400... (Integers)

x2, x4, x6, x8, x10… (Variables)

1/4, 9/25, 4/49, 81/100…(Quotients of Integers)

An expression with radicals is in _______________________________ if the following are true:

1) All radicals are broken down

2) All coefficients are reduced

3) No radicals are in the denominator

Helpful Hint: It is often easier to break down radicals first in an attempt to make the numbers more

manageable.

E1) Simplify the expression √50 P1) Simplify the expression √48

E2) Simplify the expression

a. √3

4 b.

√20

4 c. √

32

50

P2) Simplify the expression

a. √7

16 b.

√18

3 c. √

80

45

E3) Simplify the expression √8𝑤𝑥5𝑦3𝑧10 P3) Simplify the expression √150𝑥8𝑦5𝑧13

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E4) Simplify

a. 2√2 + √5 − 6√2 b. 4√3 − √27

P4) Simplify

a. 3√7 − 5√7 + 2√7 b. 8√5 + √125

E5) Simplify

a. √2 • √8 b. √2(5 − √3) c. (1 + √5)2 d. (𝑎 − √𝑏)(𝑎 + √𝑏)

P5) Simplify

a. √12 • √3 b. √3(7 + √5) c. (1 − √3)2 d. (6 − √2)(6 + √2)

E6) Simplify

a. 3

√5 b.

1

𝑐−√𝑑

P6) Simplify

a. 1

√3 b.

4

3+√2

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5.4 Complex Numbers (powers of i) (I/3)

Imaginary Numbers

The imaginary unit 𝑖, is defined as 𝑖 = √−1. The imaginary unit 𝑖 can be used to write the square root of any negative number.

* Never leave an exponent other than 1 on the imaginary unit in simplest form*

𝑖 − 𝑐ℎ𝑎𝑟𝑡 (cycle)

𝑖 = 𝑖 𝑖2 = −1 𝑖3 = −𝑖 𝑖4 = 1

Simplify:

E1. √−25 E2. √−27 E3. 3√−18 E4. √−8

3 E5. √−8 ∙ √−18 E6. (5√−6)(3√−10)

P1. √−16 P2. √−12 P3. 2√−27 P4. √−12

5 P5. √−3 ∙ √−27 P6.(3√−8)(4√−6)

Complex Numbers A complex number in standard form, where 𝑎 is the real part and 𝑏𝑖 is the imaginary part:

𝑎 + 𝑏𝑖 E7. Write the expression as a complex number in standard form

a. (4 − 𝑖) + (3 + 2𝑖) b. (7 − 5𝑖) − (1 − 5𝑖) c. 6 − (−2 + 9𝑖) + (−8 + 4𝑖)

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P7. Write the expression as a complex number in standard form

a. (−1 + 2𝑖) + (3 + 3𝑖) b. (2 − 3𝑖) − (3 − 7𝑖) c. 2𝑖 − (3 + 𝑖) + (2 − 3𝑖)

E8. Write the expression as a complex number in standard form.

a. 5𝑖(−2 + 𝑖) b. (7 − 4𝑖)(−1 + 2𝑖) c. (6 + 3𝑖)(6 − 3𝑖)

P8. Write the expression as a complex number in standard form.

a. −𝑖(3 + 𝑖) b. (2 + 3𝑖)(−6 − 2𝑖) c. (1 + 2𝑖)(1 − 2𝑖)

E9. Write the quotient 5+3𝑖

1−2𝑖 in standard form P9. Write the quotient

2−7𝑖

1+𝑖 in standard form

E10. Solve 3x2 + 10 = -26 P10. Solve 2x2 + 26 = -10

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7.1 nth Roots and Rational Exponents (I/2)

Vocabulary:

√83

813⁄

Examples: Radical Form (Rads) Rational Exponent Form (No Rads)

√4 412⁄

√𝑥23

𝑥23⁄

√8𝑥2𝑦𝑧45

815⁄ 𝑥

25⁄ 𝑦

15⁄ 𝑧

45⁄ or 2

35⁄ 𝑥

25⁄ 𝑦

15⁄ 𝑧

45⁄

√7𝑥2𝑦4𝑧6

716⁄ 𝑥

13⁄ 𝑦

23⁄ 𝑧

16⁄

Practice: Express using rational exponents (no rads)

E1. √263

E2. √36𝑥5𝑦6 E3. √𝑥610

E4. 4√2𝑎10𝑏3

Express using radical notation (rads)

E5. 𝑥35⁄ E6. 2

57⁄ 𝑎

37⁄ 𝑦

97⁄ E7. 𝑎

23⁄ 𝑔

14⁄ 𝑒

12⁄ E8. 3

23⁄ 𝑤

23⁄ 𝑚

73⁄

Simplify

E9. √325

E10. √−64𝑟6𝑤153 E11.√225𝑥3𝑦8 E12. √(2𝑥 + 1)3

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Express in simplest radical notation (rads + simplify)

E13. 𝑛74⁄ E14. 𝑥

56⁄ 𝑦

32⁄ 𝑧

73⁄ E15. √81

10 E16. (2𝑥)

23⁄ 𝑦

73⁄

Evaluate (with calculator). Round answers to the nearest hundredth (2 decimal places)

E17. √248638

E18. √953

Evaluate (without calculator). The following order might help without a calculator: 1) eliminate negative exp 2) change to radical form 3) simplify rad 4) simplify exp

E19. √643

E20. 4−1

2 E21. (√16)4 2

E22. – (25−3

2)

Solve the equation (use a calculator to approximate answers). Round answers to the nearest hundredth. The following order might help 1) isolate exponent 2) destroy exponent 3) isolate variable (solve for x) E23. 𝑥5 = 243 E24. 6𝑥3 = −1296 E25. 𝑥6 + 10 = 0 E26. (𝑥 − 4)4 = 81

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7.2 Properties of Rational Exponents (include examples with variables) (I/4)

The properties of exponents can be applied to rational exponents to simplify the expression E1. Use the properties of rational exponents to simplify the expression

a. 51

2 ∗ 51

4 b. (81

2 ∗ 51

3)2 c. (24 ∗ 34)−1

4 d. 7

713

e. (12

13

413

)

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P1. Use the properties of rational exponents to simplify the expression

a. 61

2 ∗ 61

3 b. (271

3 ∗ 61

4)2 c. (43 ∗ 23)−1

3 d. 6

634

e. (18

14

914

)

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E2. Use the properties of radicals to simplify the expression.

a. √43

∗ √163

b. √1624

√24

P2. Use the properties of radicals to simplify the expression.

a. √253

∗ √53

b. √323

√43

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E3. Write the expression in simplest form.

a. √543

b. √3

4

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P3. Write the expression in simplest form.

a. √644

b. √7

8

4

E4. Perform the indicated operation

a. 7 (61

5) + 2 (61

5) b. √163

− √23

P4. Perform the indicated operation

a. 5(43

4) − 3 (43

4) b. √813

− √33

E5. Simplify the expression. Assume all variables are positive.

a. √125𝑦63

b. (9𝑢2𝑣10)12⁄ c. √

𝑥4

𝑦8

4 d.

6𝑥𝑦12

2𝑥13𝑧−5

P5. Simplify the expression. Assume all variables are positive.

a. √27𝑦93

b. (16𝑔4ℎ2)1

2 c. √𝑥5

𝑦10

5 d.

18𝑟𝑠23

6𝑟14𝑡−3

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E6. Write the expression in simplest form. Assume all variables are positive.

a. √5𝑎5𝑏9𝑐135

b. √𝑥

𝑦73

P6. Write the expression in simplest form. Assume all variables are positive.

a. √12𝑑4𝑒9𝑓144

b. √𝑔2

ℎ7

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E7. Perform the indicated operation. Assume all variables are positive.

a. 5√𝑦 + 6√𝑦 b. 2𝑥𝑦1

3 − 7𝑥𝑦1

3 c. 3√5𝑥53

− 𝑥√40𝑥23

P7. Perform the indicated operation. Assume all variables are positive.

a. 8√𝑥 − 3√𝑥 b. 3gℎ1

4 − 6𝑔ℎ1

4 c. 2√6𝑥54

+ 𝑥√6𝑥4

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Warm-ups

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Warm-ups

Use the provided spaces to complete any warm-up problem or activity

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