GMAT Overall

Square of 14

196

Powers: Even and Odd Powers - Effect on Sign

Remember this difference between even and odd powers: An even power is always positive, whether the base is positive or negative. An odd power retains the base's original sign. Remember to determine whether or not the negative sign is included in the base: -a to n ≠ (-a) to n

Ballparking π≅3+ will become extremely useful when solving geometry questions involving circles.

Replace π with ≅3+, or " a little more than 3"

2/5

.4 40%

1/2 as a decimal and Percent

.5 50%

3/5

.6 60%

2/3

.66... 66.6...%

3/4

.75 75%

4/5

.8 80%

Square of 13

169

1/3

.33... 33.3...%

Don't of plugging in

- Don't use 0 or 1 - don't use #'s that appear in the problem or the answer choices. - Don't use the same #'s for different variables. - Don't use conversion #'s (numbers that convert between units. For example, 60 is the converstion # for minutes and hours.) Instead use a double or half of the conversion #.

Yes/No Data Sufficiency - Issue - Plug In - Plug Into Stem - DOZEN F

- Issue: Figure out the issue before diving deeper into the stmt. - Yes/No or must be: - Definite "yes" or "no" = Sufficient (S) - "maybe" = Insufficient (IS) -Plug in - Find #'s that don't contradict the stmts. - Regard the stmts as facts. - Once you find the # that completes the stmt, plug that into the ? stem. Test for Y/N. - "Is it true for any #?". DOZEN F

the only way to add or subtract powers is by extracting a common factor.

--> 2·3m·31 - 3m = 135 31 is essentially 3, so --> 2·3·3m - 3m = 135 --> 6·3m - 3m = 135 Now, extract 3m as a common factor. 3m needs to be multiplied by 6 to reach in 6∙3m, and by 1 to reach 3m: --> 3m·(6-1) = 135 --> 3m∙5 = 135 Divide by 5, and: --> 3m = 135/5 = 27

1/5

.2 20%

1/4

.25 25%

0/0 =

0/0 is known as indeterminate. But any number multiplied by 0 is 0 and so there is no number that solves the equation. In general, a single value can't be assigned to a fraction where the denominator is 0 so the value remains undefined

Square of 1

1

Exponent Equations (3 Step Process)

1) Bring both sides of the equation to the same base. 2) Ignore the bases and equate the exponents. 3) Solve

Square of 10

100

What do you plug in on % problems?

100 or its multiples (200, 300, 1000)

Square of 11

121

Square of 12

144

Square of 4

16

How to add / subtract powers with different bases?

2) Addition / Subtraction: Do not add / subtract the bases. Beware of traps: (2 to 2)+(3 to 2) ≠ 5 to 2 (7 to 2) - (4 to 2) ≠ 3 to 2

Square of 15

225

Square of 5

25

Square of 16

256

Square of 17

289

Square of 6

36

Square of 2

4

Square of 7

49

Square of 8

64

Square of 9

81

Square of 3

9

0 raised to any power (except 0) equals

= 0 0 raised to 423 = 0

1 raised to any power equals

= 1 1 raised to 3 = 1

Any number (except 0) raised to the power of 0 equals

= 1 423 raised to 0 = 1

A fractional exponent is another way of writing a ???

A fractional exponent is another way of writing a Root: The numerator of the fraction is the power. The denominator of the fraction is the root. 4√16 = 16 1/4 √4 = 4 1/2

Real Number

A real number can be anything: positive, negative, integer, fraction.

How do you multiply powers of the SAME BASE?

Add the Exponents. a to 2 times a to 3 = a to 5 a to n times a to m = a to n+m

Adding or subtracting equal roots with different bases.

Addition / Subtraction: Do not add / subtract the bases. Beware of traps: √2+√3 ≠ √5 √7 - √4 ≠ √3

An argument is made up of premises (data) and a conclusion which was drawn on the basis of these premises.

An argument is comprised of: Premise(s) - factual data or a given; Conclusion - statement, opinion, or judgment based on the premise(s). The conclusion may be logical (Example 1) or flawed (Example 2): 1. All cars have wheels. Therefore, John's car has wheels.

An even power will always be non-negative. (y) to the power of (-x) If y=-3 and x=2, then y to -x = (-3) to -2 = 1/(-3) to 2 = 1/9, which is positive

An odd power will maintain the base's original sign. (y) to the power of (-x) If y=-3 and x=1, then y to -x = (-3) to -1 = 1/(-3) to 1 = 1/-3, which is negative

Multiplying or diving equal roots with different bases.

Arithmetic operations involving equal Roots with different bases: Multiplication / Division: Combine the two bases under the same root. √2·√8 = √(2·8) = √16 = 4 √75 / √3 = √(75 / 3) = √25 = 5 Note: split complex bases into building blocks under the same roots. Look for building blocks with easily calculable roots, such as Perfect squares. √50 = √(25·2) = √25·√2 = 5·√2 Combining bases under different roots is an illogical concept. For example: √25 / ∛5 ≠ √(25/5)

How to multiply / divide powers with different bases, same exponents?

Arithmetic operations involving powers with different bases, same exponents: 1) Multiplication / Division: Combine the two bases under the same exponent. (2 to 3)·(3 to 3) = (2·3)3 = 6 to 3 15 to 3 / 3 to 3 = (15 / 3) to 3 = 5 to 3 Note: split complex bases to building blocks under the same exponents. 30 to 2 = (3·10) to 2 = (3 to 2)·(10 to 2) = 9·100 = 900

What is √2

Ballparking: Useful Numbers to Remember - √2≈1.4 When ballparking, remember: √2 is approximately 1.4. For example, 3√2 is ~3⋅1.4=4.2

How to add / subtract powers with the same base?

DON'T: add or subtract the exponents Example: (x to 3) + (x to 5) ≠ x to 8 Do: extract the highest common factor. Example: (x to 3) + (x to 5) = x to 3(1+x to 2) 2) If you're not sure that you factored the expression correctly, check that re-expanding the brackets does return to the original expression. 3) Like terms (same base and same exponent) can always be added/subtracted: (3a to 2) + (2a to 2) = 5a to 2

Difference =

Difference - the result of subtraction (-)

DOZEN F

Different One Zero Equal #'s for different variables Negative Factors

Extracting a Common Factor

Extracting a common factor is the opposite of expanding brackets. To extract a common factor: 1) Find the greatest common factor of the terms in question. The greatest common factor is the greatest integer all of the terms are divisible by. 2) Write the common factor to the left of a pair of brackets. 3) Fill in the remaining factors inside the brackets: what must the common factor be multiplied by to reach each of the terms? For example: 12+8x= The greatest common factor is 4: 4∙( ) 4 must be multiplied by 3 to receive 12, and by 2x to receive 8x: ---> 4∙(3 + 2x)

What is √9?

GMAT answer = 3 or -3

How do you find a good plug in number for problems with fractions?

Multiply the bottoms and use that number. If there are more than one number only use the number once.

Negative Exponents

Negative exponents signify a reciprocal relation. Anything raised to a negative exponent becomes 1 / [the original power without the negative sign] a to (−n) = 1 / a to n

is y to the power of (x) negative? stmt: x is even.

No. An even power will always be non-negative. plug in -3 for y and 2 for x -3 to the power of 2 = (-3)(-3) = 9 (non-negative)

is (y) to the power of (-x) positive? stmt: y is negative stmt: x is odd

No. An odd power will maintain the bases original sign. plug in -2 for y and 3 for x -2 to the power of -2 = 1/-2 to the power of 3 = 1 / (-2)(-2)(-2) = 1/-8

is (y) to the power of (-x) negative? stmt: Y is positive

No. There is no power that can turn a positive base into a negative result. plug in 2 for y and -3 for x 2 to the power of -3 = 1 / (2 to the power of 3) = 1/8 = positive

order of operations for working out any algebraic expression

Parentheses Exponents Multiplication (together with Division, from left to right) Division (together with Multiplication, from left to right) Addition (together with Subtraction, from left to right) Subtraction (together with Addition, from left to right) "Multiplication / Division" coming before "Addition / Subtraction"

Past Perfect vs Present Perfect

Past Perfect - had + V3 - actions completed before a certain point or another action: before, after, by the time, until Present Perfect - has/have + V3 (Third Form) - unknown or unspecified time, happened several times or are still occurring: for, since, already, just, yet, recently, lately, so far, ever, never, several times

Past Progressive vs Present Progressive

Past Progressive - was/were + Verb+ing, (e.g., last night at ten I was training) - an action that was in progress, when another occurred, or while one was taking place: while, as Present Progressive - am/is/are + Verb+ing, (e.g., I am eating my dinner now.) - actions that are in progress at present: now, right now, at the moment, currently, presently

Past Simple vs Present Simple

Past Simple - first verb is in V2 - occurred at a specific point (Egyptians built): yesterday, the day before yesterday, last (Friday, etc.), (an hour, etc.) ago, during (the summer, etc.) Present Simple - first verb is in V1 - actions that occur at a certain frequency: (it rains here all the time) always, all the time, usually, generally, regularly, often, sometimes, rarely, seldom, hardly ever, never, every

Perfect - Progressive - Simple

Perfect = had(has/have) + V3"ed": completed actions "has read", "had moved","had known" Progressive = "ing" taking place now/(past)- "reading" Simple = "rains, built": occurred or occurs frequently.

order of operations for working out any algebraic expression

Please Excuse My Dear Aunt Sally "Multiplication / Division" coming before "Addition / Subtraction" Solve left to right

If 3<x<6<y<10, then what is the greatest possible positive integer difference of x and y?

Plug in numbers for x and y. To get "the greatest possible...difference" they have to be as far apart as possible from each other. But who said the numbers you use have to be integers? The question asks for an integer difference, not necessarily for integer numbers. For instance x=3.5 and y=9.5. The greatest possible integer difference is 6.

base

Power is made up of a base and an exponent. x to the power of 5. (x) is the base and 5 is the exponent.

exponent

Power is made up of a base and an exponent. x to the power of 5. (x) is the base and 5 is the exponent.

Product =

Product - the result of multiplication (×)

Quotient =

Quotient - the result of division (/) Dividend÷Divisor=Quotient the quotient is the number of times the divisor divides into the dividend 10÷2=5, Dividend is 10 Divider is 2 Quotient is 5

Quotient and remainders

Quotient refers only to the integer part of the result . For instance, 7 ÷ 5=1 (2). The quotient is 1, while the remainder is 2.

Roots: Overview

Roots are simply the opposite of powers. A root asks a question: Which number, when raised to the root's power, will equal whatever is under the radical sign? For even roots, where there are two possible answers for the above question (a positive and a negative) "root" means "only the positive root". Thus, √4=2.

Scientific Notation

Scientific notation: a×10n where a (the digit term) indicates the number of significant figures in the number and 10n (the exponential term) places the decimal point. A positive exponent shows that the decimal point is shifted that number of places to the right. A negative exponent shows that the decimal point is shifted that number of places to the left. The rule: Maintain the balance between the digit term and the exponential term. If one goes up (↑) by a magnitude of 10, the other must go down (↓) by the same magnitude, and vice versa.

Subtracting with pi

Shaded regions - Subtraction with pi In the GMAT, an area is shaded when there is no geometric term to describe it directly. In such cases that area can be calculated as a subtraction of one shape from another. In the case there is a circular rim and a straight edge - The GMAT uses a subtraction involving a circle and a non-circular shape. Since does not cancel out, whenever an exact result is required you can: POE answers without POE answers without subtraction And of course- as always, POE answers out of the ballpark

Sentence Correction: Tenses - Present Progressive

Structure: am/is/are + Verb+ing, e.g., I am eating my dinner now. Usage: actions that are in progress at present Time Expressions: now, right now, at the moment, currently, presently

Sentence Correction: Tenses - Past Perfect

Structure: had + V3 (Third Form), e.g., John had moved to Seattle before he met Jane. Usage: (a) actions that had been completed before a certain point in time in the past; (b) actions that had been completed before another action in the past. Time Expressions: before, after, by the time, until

Sentence Correction: Tenses - Present Perfect

Structure: has/have + V3 (Third Form), e.g., John has had his tonsils removed. Usage: (a) actions that occurred at an unknown or unspecified time in the past; (b) actions that occurred several times in the past; (c) Actions that began in the past and are still relevant/in progress at present. Examples: John has known Jane for six years. John has not cooked dinner yet. John and Jane have lived in Seattle since 2005. Time Expressions: for, since, already, just, yet, recently, lately, so far, ever, never, several times

Sentence Correction: Tenses - Present Simple

Structure: the first verb is in V1, e.g. it rains here all the time. Usage: (a) actions that occur at a certain frequency (e.g., usually, sometimes); (b) facts and generalizations, e.g., the Earth revolves around the sun. Time Expressions: always, all the time, usually, generally, regularly, often, sometimes, rarely, seldom, hardly ever, never, every (minute, day, year, etc.), once (a day, a week, etc.)

Sentence Correction: Tenses - Past Simple

Structure: the first verb is in V2, e.g., John missed the bus to work yesterday. Usage: (a) describe historical facts: e.g., the Egyptians built the pyramids. (b) actions that occurred at a specific point in the past: e.g., John met Jane in 2005. Time Expressions: yesterday, the day before yesterday, last (Friday, etc.), (an hour, etc.) ago, during (the summer, etc.), in (835 A.D, etc.), when (followed by another sentence in the past simple tense: when he met her)

Sentence Correction: Tenses - Past Progressive

Structure: was/were + Verb+ing, e.g., last night at ten I was training. Usage: an action that was in progress (a) at a certain point in the past; (b) when another action occurred; or (c) while another action was taking place. Time Expressions: while, as (followed by Past Progressive)

How do you divide powers of the SAME BASE?

Subtract the exponents. (a) to the power of 5 divided by a to the power of 3 (a to 5) / (a to 3) = a to (5-3) = a to 2 reverse: 3 to (x-2) = 3 to x / 3 to 2 m to 3 = m to (5-2) = m to 5 / m to 2

Sum =

Sum - the result of addition (+)

Arithmetic Operations - Definitions

Sum - the result of addition (+) Difference - the result of subtraction (-) Product - the result of multiplication (×) Quotient - the result of division (:)

Averages Overview

The average of a set of numbers is a single value that describes the "center" value of the set. The average is given in the formula: Avg = Total / Number of things The average of a set of number is not necessarily part of the set. Example: The average of {3,7,8} is (3+7+8)/3= 18/3 = 6 The average of {3,4,5} is (3+4+5)/3= 12/3 = 4 Arithmetic mean, or just mean, has the same meaning as average. T = A* N

PI: Using good numbers

Use good numbers (small, positive integers) while Plugging In to make the math easy and error-free. - If the question asks about dozens of eggs, choose 24, 36, etc. - If the question asks about a number that's divisible by 15, choose a multiple of 15, etc. - If the question asks about fractions, choose a number that's a multiple of the denominators (the best way to go about it is simply to multiply the bottoms of the fractions) - If the question asks about percents, choose 100 or multiples of it.

What is √3

When ballparking, remember: √3 is approximately 1.7. so 3√3 is ~3⋅1.7=5.1.

When dividing powers of the same base, ??? the exponents.

When dividing powers of the same base, subtract the exponents. a^n / a^m = a^n-m This rule may also be applied in reverse. To do so, first rewrite the exponent as a subtraction. Next, apply the parts of the subtraction as exponents in two powers divided by one another, each with the same base as the original power. For example: x^4 = x^7-3 = x^7 / x^3 73^x = 75^x-2^x = 75^x / 72^x This is also the proper way to treat an exponent which already includes a subtraction sign: xY-5 = xy / x5 y7m-2 = y7m / y2 6x-1 = 6x / 61 = 6x / 6

Raising a power to another power you ??? the exponents.

When raising a power to another power, multiply the exponents. For example: (a^m)^n = (a^n)^m = a^m·n This can also be done in reverse. To do so, split the exponent into a product, and then 'pull' one of the multiplicands into a pair of brackets. For example, a6 can be split up to a3⋅2, which can then be rewritten as the equivalent (a3)2 or (a2)3, as needed by the question.

Raising a root to a power

When raising a root to another power, simply convert the root to power form and multiply the exponents. For example: (√a) to 2 = (a½) to 2 = a (½·2) = a to 1 = a roots and powers of the same level cancel each other out, leaving only the base. ∜a to 4 = a

Does grammer override style on sentence correction questions?

Yes. The correct answer may be stylistically flawed (i.e., redundant or ambiguous), but it can NEVER be grammatically incorrect.

Integer

an integer is simply a non-fraction or a non-decimal. Integers can be positive or negative. 0 is an integer.

Even number

an integer that is divisible by 2 with no remainder.

if √x=19 then what is x?

if the statement claimed that √x=19 (without the even power), then x would have to be 19 squared (19 to the power of 2), since 19 squared is the only number whose square root is 19.

Odd number

is not an even number

Anything raised to the power of 1 equals

itself. 423 raised to 1 = 423

When raising a power to another power...

multiply the exponents.

Non-positive and Non-negative

positive - >0 Non-negative - ≥0 Negative - <0 Non-positive - ≤0

When dividing powers of the same base, ????? the exponents

subtract the exponents. a^5 / a ^3 = a^5-3 = a^2 reverse 3^x - 2 = 3^x / 3^2

any number raised to a negative power =

the inverse. 2^-2 = 1 / (2^2) = 1/ 4

Power is made up of a base and an exponent. x to the power of 5. (x) is the base and 5 is the exponent.

which is the base? which is the exponent? (Y) to the power of (-x)

When multiplying or dividing numbers that each have a different base but the SAME EXPONENT...

you can multiply or divide the bases and keep the unique exponent. (2 to 3) times (3 to 3) = (2x3) to 3 = 6 to 3 15 to 3 divided by 3 to 3 = 15/3 to 3 = 5 to 3

zero divided by anything except zero is

zero. 0/?=0 0/0 is known as indeterminate. (for really quotient can be any number)