GMAT

X^a+b

(AB)^x

X^a-b

A^x/B^x

A^xy or (A^y)^x

1/(X)^a

^b√X^a Or ^b√(X)^a

3A<sup>x

YOU CAN ONLY ADD OR SUBTRACT NUMBERS WITH EXPONENTS IF THE BASE AND EXPONENT ARE CONSTANT</sup>

CANNOT BE SIMPLIFIED

CANNOT BE SIMPLIFIED

• 7³
• NOT 2³+3³

2³*5³ = 1000

(X+Y)(X-Y)

(x+y)(x+y) OR (x+y)²

(x-y)(x-y) OR (x-y)²

(X+1)(X-1)

+b,-a

• ZERO in the denominator means UNDEFINED
• ZERO in the numerator equals ZERO

• 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

• 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

(Last-First)/increment + 1 = # of integers

ex. (765-14)/1+1=752

ex. (24-12)/2+1=7

• 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

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

• (Y₁/X₁) = (Y₂/X₂)
• ex.
• (4/16²) = (9/v₂²)
• v₂²= 9(16²/4) = 576
• ^{√V = 24}

• Y₁X₁ = Y₂X₂
• C₁R₁ = C₂R₂
• 4*3=C₂*1
• C₂ = 2

• S₁ = 3¹ = 3
• S₂ = 3² = 9
• S₃ = 3³ = 27
• S₄ = 3⁴ = 81
• S₅ = 3⁵ = 243 65 is a multiple of 5, so it will share the same units digit of 3

1

2

3

4

5

1

8

27

64

125

~1.4

~1.7

~2.25

11

12

13

14

15

16

25

(ⁿ√x)(ⁿ√y)

CANNOT BE SIMPLIFIED FURTHER

(ⁿ√x)/(ⁿ√y)

• 4-3x<10
• -3x<6
• x>-2 REMEMBER when multiplying or dividing an inequality you need to FLIP the sign

• 1>1-ab>0
• 0>1-ab>-1
• 0<ab<1 All actions need to be done to ALL sides of the inequality

• if all the sum of the integers is divisible by 3.
• ex 72, 7+2 = 9 9/3=3

• If the integer is divisible by 2 TWICE or the last two digits are divisible by 2.
• ex. 24/2 = 12 12/2 = 6

• If the integer is divisible by BOTH 2 and 3
• ex. 90, 90/2 = 45 90/3 = 30 -> 90/6=15

• 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

• If the sum of the integers digits is divisible by 9
• ex 126 1+2+6=9 126/9=14

2

0

• 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: 2 3 5
• Factors of 24: 2 2 2 3
• Common Factors: 2 3 = 2*3 = 6

• 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

• The number LEFT OVER from long division.
• ex 3
• 5| 17
• -15
• 2 is the 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

ODD

EVEN

EVEN

EVEN

EVEN

ODD

EVEN

EVEN (also it will always be divisible by 4)

2

Can be EVEN, ODD, or NON-INTEGER

Can be EVEN, or NON-INTEGER not ODD

Can ONLY be a NON-INTEGER, not ODD not EVEN

Can ONLY be ODD or NON-Integer not EVEN

Take the average of the FIRST and LAST integer in the set

Multiply the average by the #of terms.

• Subtract the LOWEST integer from the Highest, divide by the interval b/w each integer in the set and ADD 1.
• Ex.
• Set A: 6,7,8,9,10 : 10-6=4 +1 = 5integers
• Set B: 12,14,16,18,20,22,24 : 24-12=12 /2 = 6 +1 = 7integers

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

NO - never

• 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
• zero to any power is zero

• 1
• 1 to any power is 1

• 1
• any number with 0 as the exponent is 1

• 189
• Any number to the exponent of 1 is that number

√y√y√y = y√y

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 (X²,Y³,XY,X/Y) there are usually 2 or more values for each variable.

• ex.
• (4^w)³=32d^w-1
• ((2)²)^w)³ = (2⁵)^w-1
• 6W = 5(W-1)
• W = -5

• 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

• S_n = x(kⁿ)
• K = is the constant that when multiplied by one number in the set yields the next

• S_n = x(10ⁿ)
• S₁ = 20
• 20 = x(10¹)
• x = 2

• y = mx+b
• m = constant rate of growth or decay
• b = y-intercept at time zero
• x = time

• 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

LT10 or <10

GT6 or >6 remember toe FLIP the sign when subtracting extreme values

LT16 or <16

GT(-14) remember to flip the sign when multiplying NEGATIVE extreme values

GT4 or >4 remember to FLIP the sign when DIVIDING extreme values

LT16 or <16

|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

• Avoid using 1 or 0
• Use Different numbers for each value
• Use primes
• Avoid picking numbers that appear as coefficients in the answer choices

• 'x' percent = x/100
• (x/100)y = xy/100

z-z(y/100) = z=(1-y/100)

STATED piece of information that generally provides support for a given conclusion (facts,opinions,claims-NOT the final claim/conclusion)

UNSTATED parts of the argument neccessary to reach a conclusion

• Therefore So
• As a result Consequently
• Suggests Thus
• Indicates Hence
• Accordingly It follows that

• 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
• ..at a time when
• ..on the weekend when

• BY
• ..by doing X, Y can be accomplished

• WA = weight/sum of weights(data point) + weight/sum of weights(data point)....
• ex. [2(20)+3(30)]/(2+3) = 26

To compare objects, people, or things.

To compare clauses, actions, verbs

used as in "for example"

Simply add the probability of each situation together.

P(AorB) = P(A)+P(B)-P(A+B)

Simply add the rates of each person. Only when one person is UNDOING the work do you subtract

• 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

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

Justify EVERY word in the answer choice.

Avoid answer choices with EXTREME words.

Choose teh answer choice that infers as LITTLE as possible.

• Rate * Time = Work
• Rate * Time = Distance

Add the rates of the two objects. (same if driving AWAY from each other)

Rate A - Rate B

Rate B - Rate A

~2

~3

~5

121

144

169

196

225

256

625