Number Properties

Question 1

Properties of 0

Answer 1

Integer
Neither positive nor negative
Even
Not prime

Question 2

Properties of 1

Answer 2

Integer
Positive
Odd
Not prime

Question 3

Properties of 2

Answer 3

Integer
Positive
Even
Prime

Question 4

When do you reverse the sign in an inequality?

Answer 4

When you multiply or divide by a negative number across the inequality
-(x/5) > 1
x < -5

Question 5

Multiplying and dividing fractions

Answer 5

Multiply: simply multiply across

Divide: flip second fraction and multiply

Since fraction bar indicates division, same rule applies to fractions within fractions
3/4 ÷ 9/10 = 3/4 * 10/9 = 5/6

Question 6

Reducing LARGE fractions

Answer 6

Break large fraction down into primes and then cancel

10/21 * 7/8 = (25)(7)/(37)(222) = 5/12

Question 7

Adding or subtracting fractions with common denominator

Answer 7

Only add or subtract numerator

3/4 + 2/4 = 5/4
8/11 - 3/11 = 5/11

Question 8

Adding and subtracting fractions with DIFFERENT denominators

Answer 8

Method 1: multiply each fraction by another fraction that makes both bases common. Then add or subtract

Method 2: bowtie (criss cross)
First multiply denominators to form base. The cross multiply numerators and denominators and add or subtract the products.
2/7 + 3/4 = (42) + (37) / 28 = 29/28

Question 9

2/3 - 1/2 ÷ 1/12

Answer 9

(2/3 - 1/2) = 4-3/6 = 1/6
12(1/6) = 2
2

Question 10

Order of operations

Answer 10

PEMDAS

Work exponents/roots as well as m,d,a,s from left to right

Question 11

Factors

Answer 11

numbers that can be multiplied to make another number (x is a factor of y if y is divisible by x, where x and y are integers)

1,6,2,3 are the factors of 6 (factor pairs)

Prime factors - factor tree

Question 12

A number is divisible by 2 if

Answer 12

it is even

ex. 110

Question 13

A number is divisible by 3 if

Answer 13

its digits sum to multiple of 3

ex. 18255 (1+8+2+5+5 = 21)

Question 14

A number is divisible by 4 if

Answer 14

Its last 2 digits are multiple of 4

Ex. 59456 (56÷4 = 14)

Question 15

A number is divisible by 5 if

Answer 15

it ends in a 0 or 5

Ex. 65, 70

Question 16

A number is divisible by 6 if

Answer 16

Its a multiple of both 2 and 3

Ex. 588 (even and sums to 18)

Question 17

A number is divisible by 7

Answer 17

Take LAST digit and double it. Subtract answer from remaining digits - divisible by 7 if ends in 7

Question 18

A number is divisible by 8 if

Answer 18

Its last 3 digits are multiple of 8

Ex. 17336 (336÷8 = 42)

Question 19

A number is divisible by 9 if

Answer 19

its digits add up to a multiple of 3 and 9

Ex. 6417 (6+4+1+7 = 18)

Question 20

A number is divisible by 10 if

Answer 20

it ends in a 0

Ex. 2,984,150

Question 21

A number is divisible by 12 if

Answer 21

its a multiple of both 3 and 4

Ex. 4932 (32÷4 = 8; 4+9+3+2 = 18)

Question 22

Multiples

Answer 22

numbers that can be made by multiplying by a certain number (all the numbers of an integer, y, where +/-1y, +/-2y…)

Multiples of 6: 6, 12, 18…

Question 23

Least Common Multiple

Answer 23

the least common multiple of two integers is the smallest integer that is a multiple of both

to find LCM: break down multiples into prime factors and multiply largest common primes

ex. 18 = 2 x 3^2 and 24 = 2^3 x 3
LCM: 2^3 x 3^2 = 72

Question 24

Exponent

Answer 24

exponents are just shorthand to express multiplication, and exponents can be negative (a number raised to a negative exponents is the reciprocal of that number raised to a positive exponent)

ex. x^-2 = 1/x^2

Question 25

To multiply powers with the SAME base

Answer 25

add exponents ex. 2^3 * 2^2 = 2^5

Question 26

To divide powers with the SAME base

Answer 26

subtract exponents ex. y^9/y^4 = y^5

Question 27

When exponents have different bases, you CANNOT just add or subtract

Answer 27

exs. x^4 * y^6 ≠ (xy)^10 x^5 + X^4 ≠ X^9

Question 28

A negative number raised to an EVEN power becomes positive

Answer 28

exs. ``` (-4)^2 = 16 (-1/2) = 1/4 ```

Question 29

A negative number raised to an ODD power remains negative

Answer 29

exs. ``` (-2)^3 = -8 (-1/3)^3 = -1/27 ```

Question 30

When raising a power to power, MULTIPLY the exponents

Answer 30

exs. ``` (x^3)^5 = x^15 (3^2)^3 = 3^6 = 729 ```

Question 31

When raising a product in PARENTHESES to a power, DISTRIBUTE the power over BOTH factors

Answer 31

exs. ``` (x^2*y^4)^3 = x^6*y^12 (2x^2)^4 = 2^4*x^8 = 16x^8 ```

Question 32

Any number raised to the exponent 1

Answer 32

equals itself exs. 2^1 = 2 -7^1 = -7

Question 33

1 raised to any exponent

Answer 33

equals 1 exs. 1^2 = 1 1^0 = 1

Question 34

0 raised to any power OTHER than 0

Answer 34

equals 0 exs. 0^4 = 0 0^1 = 0

Question 35

Any number raised to the 0

Answer 35

equals 1 exs. 3^0 = 1 (.34)^0 = 1 TT^0 = 1

Question 36

0 raised to the 0 power is

Answer 36

UNDEFINED

Question 37

Any positive number greater than 1 gets _____as it is raised to a higher power

Answer 37

LARGER exs. 2^2 = 4 2^3 = 8 2^4 = 16

Question 38

Any positive number less than 1 (fractions) gets _____ as it is raised to a higher power

Answer 38

SMALLER exs. (1/3)^2 = 1/9 (1/3)^3 = 1/27

Question 39

Roots can also be expressed as factional exponents

Answer 39

exs. √ x = x^1/2 4^1/2 = √4 = 2 ^3√x = x^1/3 8^1/3 = ^3√8 = 2

Question 40

Square roots with SAME base can be added or subtracted

Answer 40

exs. 4√3 + 7√3 = 11√3 5√x - 2√x = 3√x BUT 3√6 + 3√3 ≠ 3√9 (the numbers inside radicals are NOT the same)

Question 41

The product of square roots is EQUAL to the square root of the product

Answer 41

exs. ``` (√3)(√7) = √21 √721 = (√9)(√4)(√2) = 6√2 ```

Question 42

The quotient of square roots is EQUAL to the square root of the quotients

Answer 42

exs. ``` √4/√2 = 2 √63/√7 = √9 = 3 ```

Question 43

When taking the square root of a fraction, the result will always be _____ than the fraction itself

Answer 43

LARGER ex. √1/4 = 1/2 √1/4 > 1/4

Question 44

E x E

Answer 44

E

Question 45

E x O

Answer 45

E

Question 46

O x O

Answer 46

O

Question 47

E +/- E

Answer 47

E

Question 48

E +/- O

Answer 48

O

Question 49

O +/- O

Answer 49

E

Question 50

Prime numbers between 1 - 10

Answer 50

2, 3, 5, 7

Question 51

Prime numbers between 10 - 20

Answer 51

11, 13, 17, 19

Question 52

Prime numbers between 20 - 30

Answer 52

23, 29

Question 53

Prime numbers between 30 - 40

Answer 53

31, 37

Question 54

Prime numbers between 40 - 50

Answer 54

41, 43, 47

Question 55

Prime numbers between 50 - 60

Answer 55

53, 59

Question 56

Prime numbers between 60 - 70

Answer 56

61, 67

Question 57

Prime numbers between 70 - 80

Answer 57

71, 73, 79

Question 58

Prime numbers between 80 - 90

Answer 58

83, 89

Question 59

Prime numbers 90 - 100

Answer 59

97

Question 60

Absolute value

Answer 60

the distance between a number and o on the number line

Question 61

P x P

Answer 61

P

Question 62

N x N

Answer 62

P

Question 63

P x N

Answer 63

N

Question 64

Common radical approximations

Answer 64

``` TT ≃ 3 √1 ≃ 1 √2 ≃ 1.4 √3 ≃ 1.7 √4 = 2 ```

Question 65

Radical rules

Answer 65

√a√b = √ab √a/√b = √a/b a√c + b√c = (a + b)√c (√a)^2 = a BUT √a + √b ≠ √a+b √a - √b ≠ √a-b

Question 66

Exponent rules

Answer 66

``` x * x = x^2 x^-a = 1/x^a x^0 = 1 x^a*x^b = x^a+b (x^a)^b = x^ab x^a/x^b = x^a-b (negative)^odd = negative (negative)^even = positive ```

Question 67

Greatest common factor

Answer 67

largest factor that 2 numbers have in common GCF: find prime factors of both integers and combine common ones ex. 36 (2^2*3^2) and 54 (2*3^3) both have 2 and 3^2 in common so 2*3*3 = 18 = GCF

Question 68

Comparing fractions

Answer 68

need common denominator but if more than 2 fractions, compare 2 at a time use bowtie (criss-cross) method to see which is larger