Pre-requisites (Chapter P)

Question 1

Use properties of rational exponents to simplify the expression. Assume that all variables represent positive numbers.

• (243x¹⁰y²⁰)^1/5

Answer 1

3x²y⁴

Question 2

Evaluate the expression without using a calculator.

64^-7/6

Answer 2

1/2⁷

Question 3

Simplify by factoring.

³√y¹⁷

Answer 3

y⁵∛y²

Question 4

Simplify using properties of exponents.

24x^1/4
౼
4x^1/7

Answer 4

6x^3/28

Question 5

Rationalize the denominator. Simplify the answer.

7
౼
√5 + √2

Answer 5

7√5 - 7√2
౼
3

Question 6

Simplify by reducing the index number

∜-16

Answer 6

Not a real number

No number multiplied by itself 4 times can equal -16.

Question 7

Simplify by reducing the index number

∛729

Answer 7

9

Question 8

Simplify by reducing the index number

∛-64

Answer 8

-4

Question 9

Simplify using properties of exponents.

(4x^1/5)(2x^1/6)

Answer 9

8x^11/30

Question 10

Use properties of rational exponents to simplify the expression.

(4y^1/5)²
౼
y^1/15

Answer 10

16∛y

Question 11

Evaluate.

7+5(x-4)³ for x = 6

Answer 11

47

Question 12

Find the intersection

{7, 8, 9, 10, 11} ∩ {6, 8, 10, 12}

Answer 12

{8, 10}

Question 13

Find the union

{7, 8, 9, 10, 11} U {6, 8, 10, 12}

Answer 13

{6, 7, 8, 9, 10, 11, 12}

Question 14

• What are natural numbers?
• What are whole numbers?
• What are integers?
• What are rational numbers?
• What are irrational numbers?

Answer 14

• 1, 2, 3, 4, etc… (no decimals)
• 0, 1, 2, 3, 4, etc… (added a zero)
• …, -3, -2, -1, 0, 1, 2, 3, … (added negatives)
• integer/integer, repeating/terminating decimals (3/2)
• non-repeating (√2, √3)

Question 15

Consider the following set of numbers:
{-7, -3/4, 0, 0.666, √5, π, 7.3, √81}

List the numbers in the set that are:
1. Natural numbers
2. Integers
3. Irrational numbers
4. Whole numbers
5. Rational numbers

Answer 15

1. none
2. -7, √81
3. √5, π
4. 0
5. -3/4, 0.66, 7.3

Question 16

Rewrite each expression without absolute value bars.

• |√3 - 1|
• | 2 - π |
• |x|/ x if x < 0

Answer 16

• √3 - 1
• -(2-π)
• -1

Question 17

Simplify

6(2x² + 4x) + 10(4x² + 3x)

Answer 17

52x² + 54x

When the instructions say simplify, don’t factor the expression.

Question 18

Multiply each expression.

2² x 2³

Answer 18

2⁵

Question 19

Write each expression with a positive exponent.

9^-2

Answer 19

1
౼
81

Question 20

Write each expression with a positive exponent.

7x^-5y²

Answer 20

7y²
౼
x⁵

Question 21

Simplify.

(-3x⁴y⁵)³

Answer 21

-27x¹²y¹⁵

Question 22

Write each number in scientific notation.

• 34,970,000,000,000
• -0.00000000000802

Scientific notation = 1-10

Answer 22

• 3.497 x 10¹³
• -8.02 x 10¹²

Question 23

True or false: You cannot leave a radical in the denominator.

Answer 23

True

Question 24

Rationalize the denominator

7
౼
5 + √3

Answer 24

35-7√3
౼
22

Question 25

# Simplify 27^2/3

Answer 25

9

Question 26

# Simplify ⁹√x³

Answer 26

∛x

Question 27

# Factor out the greatest common factor. x(x+2)-10(x+2)

Answer 27

(x-10)(x+2)

Question 28

# Factor by grouping. x³-9x²+9x-81

Answer 28

(x²+9)(x-9)

Question 29

# Rationalize the denominator. 4 ౼ 3 - √5

Answer 29

3 + √5

Question 30

# Write in exponential form. x(⁵√)3y² ## Footnote parenthesis are there to clarify that the ⁵ is not an exponent but an index number

Answer 30

x(3y²)^1/5

Question 31

# Write in radical form. 2a^3/4b^1/4

Answer 31

2∜a³b

Question 32

# Factor the expression by grouping. 5x³-2x²-35x+14

Answer 32

(x²-7)(5x-2)

Question 33

# Factor the trinomial. 6x²-23x+21

Answer 33

(3x-7)(2x-3)

Question 34

# Factor the difference of two squares. x²-16

Answer 34

(x+4)(x-4)

Question 35

# Factor the difference of two squares. 100x²-121y²

Answer 35

(10x-11y)(10x+11y)

Question 36

# Factor the difference of two squares. x⁴-81

Answer 36

(x²+9)(x+3)(x-3)

Question 37

# Factor the following perfect square trinomial. y²+18y+81

Answer 37

(y+9)²

Question 38

# Factor the following using the formula for the sum of two cubes. y³+125

Answer 38

(y+5)(y²-5y+25)

Question 39

# Factor the following using the formula for the difference of two cubes. y³-125

Answer 39

(y-5)(y²+5y+25)

Question 40

# Factor using the formula for the sum or difference of two cubes. 64x³-1

Answer 40

(4x-1)(16x²+4x+1)

Question 41

# Factor using the formula for the sum or difference of two cubes. 8x³+27

Answer 41

(2x+3)(4x²-6x+9)

Question 42

# Factor the polynomial completely. 5x⁴-80

Answer 42

5(x-2)(x+2)(x²+4)

Question 43

# Factor the polynomial completely. x²+49

Answer 43

the polynomial is prime

Question 44

# Factor the expression completely. x³+9x²-x-9

Answer 44

(x+9)(x+1)(x-1)

Question 45

# Factor and simplify. x(x+1)^-3/4 + (x+1)^1/4

Answer 45

2x + 1 ⎼ (x+1)^3/4

Question 46

# Multiply and divide as indicated.

Answer 46

(x+6)(x+5) ౼ (x-6)

Question 47

# Subtract and provide the domains. x²+3x ౼ x²-2x-8

Answer 47

3x+6 ౼ (x-4)(x+2) ## Footnote 4, -2

Question 48

# Add and provide the domains. 3 + 4/x

Answer 48

4 + 3x ౼ x ## Footnote 0

Question 49

# Subtract and provide the domains. 6/7x - 5/3

Answer 49

18-35x ౼ 21x ## Footnote 0

Question 50

# Multiply as indicated and provide the domains. y³-27/ y²-9 * y+3/5y

Answer 50

y²+3y+9 ౼ 5y ## Footnote -3, 3, 0

Question 51

# Multiply as indicated and provide the excluded values. x-7/x-1 * x²-1/3x-21

Answer 51

x+1 ౼ 3 ## Footnote 1, 7

Question 52

# Perform the operations and provide the excluded values. x+1/x-4 - x+1/x²-7x+12

Answer 52

x+1 ౼ x-3 ## Footnote 4, 3

Question 53

# Find the LCD. x+2/2x-3 and 4/x+3

Answer 53

(2x-3)(x+3)

Question 54

# Find the LCD. 7/5x²+15x - 9/x²+6x+9

Answer 54

5x(x+3)²

Question 55

# Add or subtract. 5/6x - 4/9x²

Answer 55

15x-8 ౼ 18x²

Question 56

# Simplify. 1+1/x / 1-1/x

Answer 56

x+1 ౼ x-1

Question 57

# Perform the following operations. 3x/x²+3x-10 - 2x/x²+x-6

Answer 57

x(x-1) ౼ (x-2)(x+5)(x+3) ## Footnote 2, -5, -3

Question 58

# Perform the following operations. x/7 - 1 / x-7

Answer 58

1 ౼ 7 ## Footnote 7

Question 59

# Perform the following operations. 3x²+5x-4/x²+5x+4 - 2x/x+1 + 3/x+4

Answer 59

x-1 ౼ x+4