1 ICM Unit 0 – Algebra Rules Lesson 1 – Rules of Exponents

RULE EXAMPLE EXPLANANTION

𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛

A) 𝑥2 ∙ 𝑥6 = B) 𝑥4𝑦8𝑥3𝑦𝑧2 =

When multiplying with like bases, keep the base and add the exponents.

𝑎𝑚

𝑎𝑛= 𝑎𝑚−𝑛 𝑂𝑅

𝑎𝑛

𝑎𝑚=

1

𝑎𝑚−𝑛

𝑚 > 𝑛

A) 𝑥8

𝑥3=

B) 𝑥2𝑦5

𝑥7𝑦6=

When dividing with like bases, keep the base and subtract the exponents.

(𝑎𝑚)𝑛 = 𝑎𝑚𝑛

A) (𝑥5)3 =

Power to a Power – keep the base and multiply the exponents.

(𝑎𝑏)𝑚 = 𝑎𝑚𝑏𝑚

A) (𝑥5𝑦3)3 = B) (2𝑥3𝑧4)4 =

Power to a Product – Raise everything in the parentheses to the power.

(𝑎

𝑏)

𝑚

=𝑎𝑚

𝑏𝑚

A) (𝑥2

𝑦4)2

=

B) (3𝑥4𝑦5

4𝑥2𝑦7)3

=

Power to a Quotient – Raise everything in the parentheses to the power.

𝑎−𝑚 =1

𝑎𝑚 𝑂𝑅

1

𝑎−𝑚= 𝑎𝑚

A) 𝑥2𝑦−3

𝑧−5𝑥6=

B) 2𝑥−4

3𝑦−2 =

Change a negative exponent to a positive exponent by moving the base to either the denominator or the numerator of a fraction.

𝑏0 = 1

A) (2𝑥4)0 = B) 3𝑥0𝑦5 =

Any base raised to the zero power equals 1.

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!

2 Examples: Simplify

1. (𝑥3𝑦2)(𝑥4𝑦4𝑧)

2. (3𝑥3𝑦)(−5𝑥2𝑦2)

3. 𝑥4𝑦−5

𝑥3𝑦

4. (2𝑥−3𝑦4)

(𝑥−6)(𝑦−3)

5. (−3𝑎2𝑏2𝑐−4

4𝑎−6𝑏4𝑐−8)−2

6. −(−𝑥6)(−𝑥3)2(𝑥5)3

7. (−3𝑎3𝑏2)

3

3(𝑎𝑏4)2

8. (3𝑎2𝑏−5)

−2(2𝑎2𝑏)

4

(−6𝑎−4)−2(3𝑎−6𝑏3)2

3 ICM Unit 0 – Algebra Review Lesson 1 – Rational (Fractional) Exponents

Rule for Converting to Rational & Radical Notation

𝒙𝒂

𝒃 = √𝒙𝒂𝒃

Write each expression in simplest radical form:

1. 21

2 2. 31

2 3. 91

2 4. 251

2

5. 71

3 6. 151

4 7. x1

2 8. y−1

2

9. a2

2 10. (9a)1

2 11. (16x5)−1

2 12. 275

3

Write each expression in exponential form:

13. √7 14. √6 15. √84

16. √185

17. √x23 18.

1

√53 19. 2 ∙ √15

4 20. √(3x)7

4 Write each expression in simplest radical form – leave only one radical sign in your answer:

21. x4

7 22. 4x2

3 23. (5x)1

6

24. y2

7 ∙ y5

7 25. 32

3 ∙ b1

3

26. 41

9 ∙ x2

9 ∙ y4

9

27. 7−23 28. 3x

−1

2 29. 21

2 ∙ a2

3 ∙ b5

6

30. (x−1

2 )−4

31. x1

2( 2x1

2 + x3

2) 32. (32x10)−1

5

5 ICM Unit 0 – Algebra Review Lesson 1 Homework SHOW ALL WORK Let a and b be real numbers and let m and n be integers.

Product of Powers Property 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛 Negative Exponent Property 𝑎−𝑚 =1

𝑎𝑚

Power of a Power Property (𝑎𝑚)𝑛 = 𝑎𝑚𝑛 Zero Exponent Property 𝑎0 = 1

Power of a Product Property (𝑎𝑏)𝑚 = 𝑎𝑚𝑏𝑚 Quotient of Powers Property 𝑎𝑚

𝑎𝑛 = 𝑎𝑚−𝑛 𝑎 ≠ 0

Rational Exponent Property 𝑎𝑚𝑛 = √(𝑎𝑚)

𝑛= ( √𝑎

𝑛)

𝑚 Power of a Quotient (

𝑎

𝑏)

𝑚=

𝑎𝑚

𝑎𝑛 𝑏 ≠ 0

Evaluate the expression.

1. 42 ∙ 44

2. (5−2)3

3. 52

55

4. (3

7)

3

5. 22

2−9

6. (−9)(−9)3

Simplify the expression.

7. 𝑎6 ∙ 𝑎3

8. (𝑥5)2

9. (4𝑎2𝑏3)5

10. 𝑥8

𝑥6

11. 𝑥5

𝑥8

12. 𝑥6

𝑥6

13. (4𝑎3

2𝑏4)

2

14. (23𝑥2)5

15. (𝑥4𝑦7)−3

16. 𝑥11𝑦10

𝑥−3𝑦−1

17. −3𝑥−4𝑦0

18. 5𝑥3𝑦9

20𝑥2𝑦−2

19. 𝑥5

𝑥−2

20. 𝑥5𝑦2

𝑥4𝑦0

21. (𝑥3)0

22. (10𝑥5𝑦3)−3

23. 𝑥−1𝑦

𝑥𝑦−2

24. (4𝑥2𝑦5)−2

25. 2𝑥2𝑦

6𝑥𝑦−1

26. 𝑥𝑦9

3𝑦−2∙

−7𝑦

21𝑥5

27. 18𝑥𝑦

7𝑥4∙

7𝑥5𝑦2

4𝑦

6 ICM Unit 0 – Review of Algebra Lesson 2 – Factoring I. Greatest Common Factor (GCF) → if possible, always do this FIRST.

A. 24𝑎2𝑏 − 18𝑎𝑏2

B. 5𝑥2𝑦 − 20𝑥𝑦2𝑧 + 35𝑦3𝑧2

C. 2𝑥3𝑦𝑧3 − 7𝑥𝑦5𝑧2

II. Difference of Squares Factoring → 𝒂𝟐 − 𝒃𝟐 = (𝒂 − 𝒃)(𝒂 + 𝒃) *** Always check for a GCF first!!!!

A. 𝑥2 − 9

B. 𝑥2 − 49

C. 𝑥2 − 36𝑦2

D. 16𝑥2 − 1

E. 𝑥2 + 25

F. −1 + 𝑥2

G. 24𝑥5 − 54𝑥𝑦6

H. 4𝑥2 − 64

I. 𝑥4 − 16

III. Sum and Difference of Cubes →SUM 𝑎3 + 𝑏3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2)

→DIFFERENCE 𝑎3 − 𝑏3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏2)

*** Always check for a GCF first!!!!

A. 𝑥3 − 125

B. 27𝑥3 − 1 C. 64𝑥3 − 8𝑦3

D. 32𝑥3 − 500

E. −3𝑥3 + 192𝑦3 F. 𝑥3 + 1

G. −32𝑥3 − 4𝑦3

H. 448𝑥3 + 189 I. 8𝑥6 + 27𝑦3

7

IV. Factoring Trinomials → 𝑥2 + 𝑏𝑥 + 𝑐 *** Always check for a GCF first!!!!

A. 𝑥2 + 9𝑥 + 20

B. 𝑥2 − 7𝑥 + 10

C. 𝑥2 + 3𝑥 − 40

D. 𝑥2 − 3𝑥 − 10

E. 2𝑥2 − 8𝑥 − 90

F. 𝑥4 − 7𝑥2 + 12

V. Factoring Trinomials → 𝒂 𝑥2 + 𝑏𝑥 + 𝑐 *** Always check for a GCF first!!!!

A. 2𝑥2 + 7𝑥 + 6

B. 2𝑥2 − 9𝑥 + 4

C. 3𝑥2 + 5𝑥 + 2

D. 6𝑥2 − 4𝑥 − 42

E. 6𝑥2 + 11𝑥𝑦 + 4𝑦2

F. 5𝑥4 − 17𝑥2 + 14

VI. Factoring by Grouping 1. Check for GCF 2. Group 3. GCF of each group 4. Binomial GCF.

A. 𝑥3 − 5𝑥2 + 3𝑥 − 15

B. 4𝑥2 + 20𝑥 − 3𝑥𝑦 − 15𝑦

C. 3𝑥3 − 6𝑥2 + 15𝑥 − 30

D. 𝑥2 + 𝑎𝑏 − 𝑎𝑥 − 𝑏𝑥

E. 𝑥3 + 2𝑥2 − 9𝑥 − 18

F. 9𝑥3 + 36𝑥2 − 4𝑥 − 16

8 ICM Unit 0 – Algebra Review Lesson 3 – Simplifying Rational Expressions

Rational Function: Quotient of 2 polynomials written in simplest form with a denominator that cannot

be equal to zero (𝑬𝒙: 𝒙+𝟐

(𝒙−𝟏)(𝒙+𝟑); 𝒙 ≠ 𝟏 𝒐𝒓 − 𝟑)

A rational function is undefined when the denominator is equal to zero.

1) Factor the denominator completely. 2) Set each factor equal to zero and solve. 3) State what values make the function undefined.

Example: For what values of 𝑥 is the following function undefined? 𝑓(𝑥) =𝑥2−3𝑥+2

𝑥3−4𝑥

1) Factor the denominator completely: 2) Set each factor equal to zero and solve: 3) State the undefined values: The function 𝒇(𝒙) will be undefined at 𝒙 =

**From this point on, we will assume that the replacement set of the variables in the fraction includes no numbers for which the denominator will be equal to zero.

Simplest Form: Answers to all fraction problems should be in simplest form. To put rational expressions in simplest form, factor everything completely (numerator and denominator) and then divide out (cross out) the common factors in the numerator and denominator.

Simplify:

1. 𝑥2+4𝑥

𝑥2−16

2. 𝑥2−9

(𝑥+3)2

3. 3𝑥3−𝑥4

2𝑥3−6𝑥2

4. 𝑥2−4𝑥−5

𝑥2−5𝑥−6

5. 𝑥5−3𝑥4−4𝑥3

𝑥3−6𝑥2+8𝑥

6. 𝑥2−8𝑥+16

4−𝑥

9 Adding and Subtracting Rational Expressions Simplify:

1. 2

3+

3

5

2. 7𝑥

12𝑦2 +4𝑦

6𝑥2

3. 4

𝑥+5+

3𝑥+7

2𝑥+10

4. −5𝑦

𝑦2−9+

𝑦

𝑦2+3𝑦

5. 2𝑥

𝑥2−25−

𝑥

𝑥2−10𝑥+25

6. 𝑥−8

𝑥2−6𝑥+8−

𝑥+6

𝑥2+𝑥−6

10 Multiplying and Dividing Rational Expressions Simplify:

1. 4𝑎

5𝑏∙

15𝑏

16𝑎

2. 7𝑎

9𝑏∙

63𝑏3

35𝑎2

3. 4𝑥2𝑦

15𝑎3𝑏3÷

2𝑥𝑦2

5𝑎𝑏3

4. 𝑥+2

𝑥+3÷

𝑥2+𝑥−12

𝑥2−9

5. 𝑥2−3𝑥−4

𝑥3 ∙𝑥2+2𝑥

2𝑥−8

6. 𝑥−2

𝑥2+5𝑥∙

𝑥2−25

2𝑥−𝑥2

Multiplying Polynomials Simplify:

1. (3𝑥4𝑦 + 4𝑥𝑦3 − 12𝑥 − 5𝑦 + 4)(𝑥2𝑦𝑧2)

2. (2𝑥2𝑦3 − 𝑧3)(−𝑥𝑧8 − 𝑥𝑦4𝑧2)

11 ICM Unit 0 – Algebra Review Lesson 3 Homework SHOW ALL WORK Simplify

1. 74

36

56

24

yx

yx 2.

105

862

x

xx 3.

152

4032

2

xx

xx

Multiply or divide (remember to factor when necessary).

4. 34

2

45

1862

2

2

xx

xx

xx

x 5.

153

20

124

12 22

x

xx

x

xx

6. 35

45

5

9

4

3

6

2

x

x

x

x 7.

6 24

9 20

5 25

3 62

x

x x

x

x

8. 7 14

4

12

2 42

3x

x

x

x

9.

3 21

3 28

5 20

2 82

x

x x

x

x

10. x x

x

x x

x

2 27 8

2 6

3 4

4 12

11.

25

35

14

10

3

4 2 2 3

xy

x y

xy

x y

Add or Subtract.

1. 4

1

8

3

x 2.

5

7

306

2

xx

3. x

x

3

4

12

7 4.

158

2

3

32

xy

y

y

5. 284

2

7

5

x

x

x

x 6.

2411

3

8

62

xy

y

y

12 ICM Unit 0 – Algebra Review Lesson 4 – Dividing Polynomials

13

14 ICM Unit 0 – Algebra Review Lesson 4 Homework SHOW ALL WORK

15 ICM Unit 0 – Algebra Review Lesson 5 – Radical Expressions

16

17

18

(√3 + √5)(√2 + 4√3)

5√6 − 3√24 + √150

19 ICM Unit 0 – Algebra Review Lesson 5 Homework SHOW ALL WORK

20

21 ICM Unit 0 – Algebra Review Lesson 6 – Solving Radical Equations

There are four steps to solving a radical equation: 1) Isolate the radical. 2) Raise both sides to the power of the root. 3) Solve for x. 4) Check for extraneous solution(s). What is an EXTRANEOUS solution? A solution to the final equation but not to the original equation. Extraneous solutions can occur when solving a square root equation but not when solving linear, quadratic or exponential equations. Examples:

1. √𝑥 = 8 2. √𝑥 + 7 = 8 3. 2√𝑥 + 6 = 14

4. −4√𝑥 + 11 = 3 5. (𝑥 − 2)1

2 − 2 = 2 6. 10 − 3√2𝑥 + 53

= −11

22

7. √10𝑥2 − 49 = 3𝑥 8. √2𝑥 − 6 = √5𝑥 − 15 9. √6𝑥 − 53

= √3𝑥 + 23

10. √3𝑥 + 7 = 𝑥 + 1 11. √15 − 7𝑥 = 𝑥 − 1

12. √𝑥 + 2 = 4 − √𝑥 13. √𝑥 + 3 = √𝑥 + 4

14. √𝑥 + 8 = √𝑥 + √3 15. √𝑥 + 3 = √𝑥 + 1 + 1

23 ICM Unit 0 – Algebra Review Lesson 6 Homework

1. √𝑥 − 1 = 3 2. 2 = √

𝑥

2

3. √−8 − 2𝑥 = 0 4. (𝑥 + 4)1

2 = 7

5. √𝑥 − 33

= 5 6. √2𝑥 − 6 = √3𝑥 − 14

7. √8𝑥 = 𝑥 8. √9 − 𝑥3

= √1 − 9𝑥3

9. √3 − 2𝑥 = √1 − 3𝑥 10. 𝑥 = (20 − 𝑥)1

2

24 ICM Unit 0 – Algebra Review Lesson 7 – Solving Rational Equations When solving equations with variables in the denominator, you must check the solution to be sure the denominator will not

equal zero. The solution will be eliminated if the denominator is zero.

Solve:

1. 4

3=

4𝑥+6

5𝑥−3

2. 9

14+

3

𝑥+2=

3

4

3. 4

𝑥−2+

9

𝑥2−4=

−2

𝑥+2

4. 5𝑥

𝑥+1+

2

𝑥=

5

1

5. 𝑥

2−

9−2𝑥

𝑥−7=

5

𝑥−7

6. 𝑥

𝑥+2−

𝑥+2

𝑥−2=

𝑥+3

𝑥−2

25 ICM Unit 0 – Algebra Review Lesson 7 Homework SHOW ALL WORK