3A^{x
YOU CAN ONLY ADD OR SUBTRACT NUMBERS WITH EXPONENTS IF THE BASE AND EXPONENT ARE CONSTANT }
A^{x}+B^{y}
CANNOT BE SIMPLIFIED
AB^{x}
CANNOT BE SIMPLIFIED
Simplify:
(2+5)^{3}
7^{3}
NOT 2^{3}+3^{3}
Simplify:
(2*5)^{3}
2^{3}*5^{3} = 1000
X^{2}-Y^{2}
(X+Y)(X-Y)
X^{2}+2XY+Y^{2}
(x+y)(x+y) OR (x+y)^{2}
x^{2}-2xy+y^{2}
(x-y)(x-y) OR (x-y)^{2}
X^{2}-1
(X+1)(X-1)
What are the Possible values for x ?
(x-b)(x+a)/(x-m) = 0
+b,-a
ZERO in the denominator means UNDEFINED
ZERO in the numerator equals ZERO
How do you calculate the SUM of a range of integers?
ex. 20->100
Multiply the middle term(average) by the number of terms
ex.
Average = (100+20)/2 = 60
#of Terms = (100-20)+1 = 81
Sum = 60*81 = 4860
For any set where there is an ODD number of terms, the sum of all the integers is ALWAYS a multiple of the # of items (NOT TRUE if there is an Even number of terms) ex. 4860/81=60
What is the SUM of an EVENLY spaced Set?
ex. 4,8,12,16,20
The sum of an evenly spaced set is the average times the number of terms
ex.
Average = (20+4)/2 = 5
#of terms = (20-4)/4+1 = 5 *after subtracting the highest and lowest digits, you need to divide by the increment b/w digits in the set before adding 1
Sum = 12*5 = 60
How many integers are there in a consecutive set?
ex. 14->765inclusive
(Last-First)/increment + 1 = # of integers
ex. (765-14)/1+1=752
How many integers are there in a consecutive set?
ex. all even integers 12 to 24 inclusive
ex. (24-12)/2+1=7
How many integers are there in a consecutive set?
ex. multiples of 7 between 100 and 150inclusive
ex. (147-105)/7+1=7
Need to use the lowest and highest multiple of 7 within that range
21*7 = 147 closest integer to 150 without going outside the range
15*7 = 105 closest integer to 100 without going outside the range
What is the Factor Foundation Rule?
If A is a factor of B, and B is a factor of C, then A is also a factor of C.
The product of any combination of factors of integer N is in an of itself a factor of integer N
What is the forumal for DIRECT proportionality?
ex. Height is the square of the velocity of an object, if an object is moving at 16ft/s reaches 4ft, what speed is required to reach 9ft?
(Y_{1}/X_{1}) = (Y_{2}/X_{2})
ex.
(4/16^{2}) = (9/v_{2}^{2})
v_{2}^{2}= 9(16^{2}/4) = 576
^{√V = 24}
What is the formula for INVERSE proportionality?
ex. Current is inversley proportional to resistance, if a wire carries 4amps of current but the resistance decreases by 1/3 of its original value how many amps of current will flow?
Y_{1}X_{1} = Y_{2}X_{2}
C_{1}R_{1} = C_{2}R_{2}
4*3=C_{2}*1
C_{2} = 2 _{}
Use patterns when finding terms in a sequence.
ex. What is the units digit of S_{65} if S_{n} = 3^{n}
S_{1} = 3^{1} = 3
S_{2} = 3^{2} = 9
S_{3} = 3^{3} = 27
S_{4} = 3^{4} = 81
S_{5} = 3^{5} = 243 65 is a multiple of 5, so it will share the same units digit of 3
^{3}√1
1
^{3}√8
2
^{3}√27
3
^{3}√64
4
^{3}√125
5
1^{3}
1
2^{3}
8
3^{3}
27
4^{3}
64
5^{3}
125
√2
~1.4
√3
~1.7
√5
~2.25
√121
11
√144
12
√169
13
√196
14
√225
15
√256
16
√625
25
^{n}√xy
(^{n}√x)(^{n}√y)
^{n}√(x+y)
CANNOT BE SIMPLIFIED FURTHER
^{n}√(x/y)
(^{n}√x)/(^{n}√y)
Solve 4-3x<10
4-3x<10
-3x<6
x>-2 REMEMBER when multiplying or dividing an inequality you need to FLIP the sign
Solve for ab 1>1-ab>0
1>1-ab>0
0>1-ab>-1
0<ab<1 All actions need to be done to ALL sides of the inequality
What is the test to see if an integer is divisible by 3?
if all the sum of the integers is divisible by 3.
ex 72, 7+2 = 9 9/3=3
What is the test to see if an integer is divisible by 4?
If the integer is divisible by 2 TWICE or the last two digits are divisible by 2.
ex. 24/2 = 12 12/2 = 6
What is the test to see if an integer is divisible by 6?
If the integer is divisible by BOTH 2 and 3
ex. 90, 90/2 = 45 90/3 = 30 -> 90/6=15
What is the test to see if an integer is divisible by 8?
If the integer is divisible by 2 THREE times or the last 3 digits are divisible by 8.
ex118 128/2= 64 64/2 = 32 32/2=16 128/8=16
What is the test to see if an integer is divisible by 9?
If the sum of the integers digits is divisible by 9
ex 126 1+2+6=9 126/9=14
What is the lowest PRIME number?
2
What number is a MULTIPLE(more) of EVERY number?
0
What is the Greatest Common Factor (GCF)?
The largest divisor of two or more integers. Calculated by multiplying ONLY the common factors a set of integers has.
Ex. GCF of 30 and 24
Factors of 30: 23 5
Factors of 24: 2 2 2 3
Common Factors: 2 3 = 2*3 = 6
What is the Least Common Multiple (LCM)?
The smallest multiple of two or more integers. Calculated by multiplying all the primes of each integer but ONLY counting the common factors ONCE.
Ex. LCM of 30 and 24
Factors of 30: 2 3 5
Factors of 24: 2 2 2 3
Multiply Factors: 2*2*2*3*5 = 120
What is a remainder?
The number LEFT OVER from long division.
ex 3
5| 17
-15
2 is the remainder
How can you pick a number with a desired remainder?
If you need a desired remainder of N after division by X, pick a multiple of X and ADD N
ex. Need a number with a remainder of 4 after division by 7,
14+4=18
2
7| 18
-14
4 is the remainder
ODD+EVEN = ?
ODD
ODD-EVEN = ?
ODD
ODD+ODD = ?
EVEN
ODD-ODD = ?
EVEN
EVEN-EVEN = ?
EVEN
EVEN+EVEN = ?
EVEN
ODD*ODD = ?
ODD
ODD*EVEN = ?
EVEN
EVEN*EVEN = ?
EVEN (also it will always be divisible by 4)
If the sum of 2 primes is ODD, what must one of the primes be?
2
EVEN/EVEN = ?
Can be EVEN, ODD, or NON-INTEGER
EVEN/ODD = ?
Can be EVEN, or NON-INTEGER not ODD
ODD/EVEN
Can ONLY be a NON-INTEGER, not ODD not EVEN
ODD/ODD = ?
Can ONLY be ODD or NON-Integer not EVEN
How can you get the average of an evenly spaced set of integers?
Take the average of the FIRST and LAST integer in the set
How do you calculate the SUM of the terms in an evenly spaced set?
Multiply the average by the #of terms.
How can you find the #of terms between two integers?
Subtract the LOWEST integer from the Highest, divide by the interval b/w each integer in the set and ADD 1.
If you have an ODD number of consecutive integers, will the average be an integer?
YES - always
The sum of an ODD number of consecutive integers will also ALWAYS be a multiple of the number of terms
ex. 4+5+6+7+8 = 30, a multiple of 5
If you have an EVEN number of consecutive integers, will the average be an integer?
NO - never
How do you calculate the sum of different parts of a set?
Use N as the first integer and add the interval to each consecutive integer starting with N.
ex. How much greater is the sum of the last four integers in a set of eight than the first 4
(n+4)+(n+5)+(n+6)+(n+7)
-[ n +(n+1)+(n+2)+(n+3)]
4 + 4 + 4 + 4 = 16 , the last 4 intergers is 16 more than the first 4 integers
0^{3}
0
zero to any power is zero
1^{6}
1
1 to any power is 1
17^{0}
1
any number with 0 as the exponent is 1
189^{1}
189
Any number to the exponent of 1 is that number
√(y^{3})
√y√y√y = y√y
REMEMBER variables with EVEN exponents always have TWO values, a negative and positive
Variables with ODD exponents have ONE value, same sign as the base
What is the MASTER RULE?
If two equations are LINEAR (no squared terms) the equations WILL be sufficient to find the value of the variables unless the two equations are mathematically identical ex. x+y=12 2x+2y=24
if there are ANY NON-linear terms (X^{2},Y^{3},XY,X/Y) there are usually 2 or more values for each variable.
Same base means same exponent
ex.
(4^{w})^{3}=32d^{w-1}
((2)^{2})^{w})^{3} = (2^{5})^{w-1}
6W = 5(W-1)
W = -5
^{}
What is the LINEAR FORMULA used for determining a value in a sequence?
ex What is the rule for the sequence: 16,20,24,28
S_{n} = k_{n}+x
K = the constant interval between integers in a set
Sn = 4n+x
S1 = 4(1)+x
16 = 4+x
x = 12
What is the EXPONENTIAL FORMULA used for determining a value in a sequence?ex What is the rule for the sequence: 20,200,2000,20000
S_{n} = x(k^{n})
K = is the constant that when multiplied by one number in the set yields the next
S_{n} = x(10^{n})
S_{1} = 20
20 = x(10^{1})
x = 2
What is the formulat for LINEAR GROWTH
y = mx+b
m = constant rate of growth or decay
b = y-intercept at time zero
x = time
How is Q related to each variable in the below formula:
Q = 4W/4xz^{2}
Q is DIRECTLY proportional to W, if one increases so does the other
Q is Inderectly proportional to x and z, if one increases the other decreases
NEVER SUBTRACT OR DIVIDE INEQUALITIES, you can combine through multiplication or addition
8+LT2
LT10 or <10
8-LT2
GT6 or >6 remember toe FLIP the sign when subtracting extreme values
8*LT2
LT16 or <16
-7*LT2
GT(-14) remember to flip the sign when multiplying NEGATIVE extreme values
8/LT2
GT4 or >4 remember to FLIP the sign when DIVIDING extreme values
LT2*LT8
LT16 or <16
What is the value(s) for X in |x-2|<5?
|x-2|<5
absolute value formulas have 2 possible values, when the absolute value is negative or positive
x-2<5 -(x-2)<5
x<7 -x+2<5
x>-3
-3<x<7 is the range
What are the rules for picking numbers?
Avoid using 1 or 0
Use Different numbers for each value
Use primes
Avoid picking numbers that appear as coefficients in the answer choices
What is the formula for 'x' percent of y?
'x' percent = x/100
(x/100)y = xy/100
What is the formula for 'y' percent less than z?
z-z(y/100) = z=(1-y/100)
What is a Premise?
STATED piece of information that generally provides support for a given conclusion (facts,opinions,claims-NOT the final claim/conclusion)
What is an Assumption?
UNSTATED parts of the argument neccessary to reach a conclusion
What are the basic CONCLUSION signal words.
Therefore So
As a result Consequently
Suggests Thus
Indicates Hence
Accordingly It follows that
What are the rules for determining a claim?
1. Predict the future; will,should,can expect to, are likely to
2. Subjective Opinion
3. Cause and Effect; if there is more than one claim which claim leads to the other(final claim)
X therefore Y test
When describing/using a period in time, what relative pronoun must be used WHICH or WHEN?
WHEN
..at a time when
..on the weekend when
What preposition needs to be used when explaining HOW something is done?
BY
..by doing X, Y can be accomplished
What is the formula for WEIGHTED AVERAGE?
ex. What is the WA of two 20s and 3 30s
WA = weight/sum of weights(data point) + weight/sum of weights(data point)....
ex. [2(20)+3(30)]/(2+3) = 26
When is LIKE used?
To compare objects, people, or things.
When is AS used?
To compare clauses, actions, verbs
When is SUCH AS used
used as in "for example"
If two situations CANNOT occur together, what is the probability of one OR the other occurring separately?
Simply add the probability of each situation together.
If two situations CAN occur together, what is the probability of ONE happening and NOT the other?
P(AorB) = P(A)+P(B)-P(A+B)
How do you calculate the COMBINED work of people WORKING TOGETHER?
Simply add the rates of each person. Only when one person is UNDOING the work do you subtract
How do you calculate average rate?
If given the individual rates and distances, calculate each individual time and add.
Rate * Time = Distance
going 4mph 3 12
returning 6mph 2 24
4.8 5 24
If ONLY given the rates, PICK distances that have a multiple of those rates
When "WHICH" follows a comma, the word "WHICH" is referring to is the subject closest to it before the comma.
We like the store, which is large. CORRECT
We liked the rugs in the store, which were fur INCORRECTLY STATING STORE IS MADE OF FUR
What are the 3 rules of READING COMPREHENSION?
Justify EVERY word in the answer choice.
Avoid answer choices with EXTREME words.
Choose teh answer choice that infers as LITTLE as possible.
What is the formula for RATE?
Rate * Time = Work
Rate * Time = Distance
How do you combine the rates of 2 objects that are driving TOWARD each other?
Add the rates of the two objects. (same if driving AWAY from each other)
At what rate with CAR A catch up to CAR B?
Rate A - Rate B
How do you calculate the rate of Car A if it is chasing Car B and falling behind?