Multiplication & Roots
21²
441 12²=144 21²=441
45-45-90 triangles
45-45-90 The base and height are equal each leg = x hypotenuse = x√2
9x8
72
Is 1 a prime number?
No
3.102 x 10² in normal form
shift the decimal 2 places to the right (because 10²) 3.102 → 310.2
Surface area of cylinder
2x the surface area of the circle, then the rectangle in the middle. SA = 2πr² + (circumference of circle x h)
π
3.14 this is actually the ratio of the circumference to the diameter. ie: every circle's circumference is 3.14x its diameter
30-60-90 triangles
30-60-90 30° leg = x 60° leg = x√3 90° leg = 2x
3x 3x 3x equals
3³x³ which is the same thing as 3x³
√50²
50
2⁹
512
23²
529
6x9
54
9x6
54
7x8
56
5⁶ ÷ 5⁴
5² subtract the exponents
8x9
72
2³
8
3⁴
81
9x9
81
12x7
84
division algorithm
A divided by B A = BQ + R Q = quotient R = remainder
√225
15
2⁴
16
8x2
16
√256
16
13²
169
√289
17
√324
18
14²
196
isosceles
2 sides of equal length
√5
2.25
√441
21 12²=144 21²=441
√484
22
15²
225
√529
23
√576
24
3⁵
243
√625
25
16²
256
2⁸
256
3x9
27
3³
27
4x7
28
7x4
28
canonical forms - even
2n
canonical forms - odds
2n + 1
probability
# of possibilities that meet my conditions ÷ # of possibilities
Sum of interior angles of a polygon
(#sides - 2) * 180°
if your average free throw percentage is 75%, what is the likelihood of getting 10 free throws in a row?
(.75)¹⁰
2(3ⁿ+3ⁿ+3ⁿ)=162
(3ⁿ+3ⁿ+3ⁿ) = 3(3ⁿ) = 3ⁿ⁺¹ 2(3ⁿ⁺¹)=2¹3⁴
6³ x 6⁶
(6x6x6)(6x6x6x6x6x6) = 6⁹ so you add the exponents
(a³)⁴
(axaxa)(axaxa)(axaxa)(axaxa) total of 12 a's = a ¹² so here you multiply the exponents
area of trapezoid
(b₁+b₂)h ÷ 2
10-³
.001 here the 3 represents number of PLACES behind the decimal, NOT zeroes
if n is a positive integer and the product of all the integers from 1 to n is divisible by 990, what is the least possible value of n?
1 to n = n! every prime factor of 990 must be a factor of n! so... prime factor 990.. pick the greatest number
10³
1,000 1 with 3 zeroes
√2
1.4 Valentine's day
√3
1.7 St. Patricks Day
area of triangle
1/2 b x h
2-⁴
1/2⁴ = 1/16
x-¹
1/x
2¹⁰
1024
12x9
108
9x12
108
11x11
121
5³
125
2⁷
128
√169
13
11x12
132
12x11
132
probability - number of possibilities
you are rolling two die - how many possibilities of combinations are there? 6 x 6 = 36
rate problems - formula
D/W = R × T
Divisibility rules: 8
If the last 3 digits are divisible by 8
probability of a OR b
P(a) + P(b) - P(a&b)
permutations vs. combinations contd
a combination is when order DOESN'T matter. For example, you have 5 shirts, but only space for 3 shirts in your suitcase. To solve: 5 × 4 × 3 ÷ 3 × 2 × 1
permutations vs. combinations
a permutations is when order DOES matter. For example, when you have 3 seats available and 5 people. To solve: 5 × 4 × 3
difference of squares
a² - b² = (a + b)(a - b)
a² x aⁿ
a²+ⁿ we learned in the card above that we add exponents with equal bases.
area of parallelogram
b x h
rate problems - combined rates
combined rates is just the sum of the rates
.0000516 in scientific notation
count digits up to the first non-zero: 5 5.16 x 10-⁵
7,012,000,000,000 in scientific notation
count the digits after the first non-zero (in this instance the number of digits after the 7): 12 7.012 x 10¹²
number line: |a+b|
distance between a and -b on the # line ex: |x+1| ≤ 4 -1 would be the center, with 4 on either side so.. -5 to 3
number line: |a-b|
distance between a and b on the # line ex: |x-1| ≤ 4 +1 would be the center, with 4 on either side so.. -3 to 5
average speed is NOT the average of the speeds
it is total distance ÷ total time
for a prime to divide a factorial
it must be in the factorial
triangle inequality theorem
length of a side must be < sum of lengths of other 2 sides
multiples/factors
more multiples, fewer factors.
likelihood of a coin landing on heads and an even rolling a 6-sided die
multiply each likelihood = 1/2 x 3/6 = 1/2 x 1/2 = 1/4
probability of getting heads, tails, heads, in this exact order P (H,T,H)
multiply probability of each 1/2 x 1/2 x 1/2 = 1/8
algebraic expression for: the sum of {1,2,3,4....n)
n(n+1)/2 Ex: The sum of {1,2,3,4} 4(4+1)/2 4(5)/2 20/2 = 10
algebraic expression for: the sum of {12,14,16,18....n} where there are 250 terms
n₁+n₂₅₀ ÷ 2 × 250
must be =
process of elimination
1.75 x 10-³ in normal form
shift the decimal 3 places to the left 1.75 →.00175
Volume of a cylinder
surface area of the circle, times the height V = πr²h
dividend
the number being divided ie: 8 divided by 2 dividend = 8
divisor
the number you are dividing the main number by "x divided by DIVISOR" ie: 8 divided by 2 divisor = 2
rate problems - if work isn't given
then W = 1
circumference =
πd where d = diameter
√196
14
12²
144
9x2
18
√361
19
17²
289
2⁵
32
4x8
32
8x4
32
18²
324
7³
343
3x12
36
4x9
36
9x4
36
19²
361
√2⁴
4
6x7
42
4x12
48
6x8
48
8x6
48
22²
484
7²
49
8x7
56
24²
576
12x5
60
25²
625
5⁴
625
7x9
63
9x7
63
2⁶
64
12x6
72
6x12
72
7x12
84
3²
9
12x8
96
8x12
96
Always chose which answers to check
B & D
Divisibility rules: 9
If the sum of the digits is divisible by 9
Is 2 a prime number?
Yes, the only even prime
Divisibility rules: 6
divisible by 2 and 3
0 is even or odd
even
even * even
even
even * odd
even
even + even
even
odd + odd
even
Divisibility rules: 7
if the diff between 2x the ones digit & the remaining digits is a multiple of 7 ex: 672 → 67-2(2) = 63 = a multiple of 7 (chop the ones digit)
Divisibility rules: 11
if the diff between the sum of the digits in odd places & even places (left to right) is a multiple of 11 ex: 2,849→ (9+8) - (4+2) = 11-11 = 0 → counts as divisible by 11
Divisibility rules: 4
if the last 2 digits are divisible by 4
Divisibility rules: 3
if the sum of the digits is divisible by 3
even + odd
odd
odd * odd
odd
Powers of 2 patterns (2ⁿ)
units digit of answer: 2,4,8,6,2,4,8,6
√x²
|x|
