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Algebra -

-Solving for one or more unknown

-Many times not providing a solution to a problem, Rather its simply manipulating an equation so you can recognize an equivalent(equal) form

-Roots & Exponents

-Many times not providing a solution to a problem, Rather its simply manipulating an equation so you can recognize an equivalent(equal) form

-Roots & Exponents

x^5 =

= x*x*x**x****x***x*x*x*x

x^1 =

x

x^0 =

1

x^ -a =

1 / x^a

1 / x^ -a =

x^a

Rule: x^a * x^b =

x^a+b

YOU CAN only add exponents when bases are IDENTICAL!!!

y^a ** x^b = (x**y)^a+b

b/c y & x are different you can't just a+b to = c

b/c y & x are different you can't just a+b to = c

Rule: x ** x^a ** x^a =

X ^ 1+a+a = X^1+2a

x ^a / x^b =

x^a-b ; Same as x^a * x^-b

ex. 4^10 / 4^7 = 4^3

ex. 4^10 / 4^7 = 4^3

Key: Don't forget about the implied 1

ex. X=X^1 or X / X^a = X^1-a

Rule: (x^a)^b =

x^a*b

ex. (7^3)^8 = 7^24

ex. (7^3)^8 = 7^24

X^(3)^2 =

x^9 NOT x^6

4^4 ** 2^3 ** 8^2 =

(2^2)^4 ** 2^3 ** (4*2)^2 =

2^8** 2^3 ** (2*2^2)^2 =

2^8** 2^3 ** (2^1+2 = 2^3)^2 =

2^8+3 * 2^6 = 2^11+6 = 2^17

2^8

2^8

2^8+3 * 2^6 = 2^11+6 = 2^17

ex. 27^(2x+4) = 3^(3x+9); X=?

(3^3)^(2x+4) = 3^6x+12=3^3x+9

3x=-3 ; x=-1

3x=-3 ; x=-1

Rule: (x*y)^a =

x^a * y^a

ex. (3**4)^2 = 3^2 ** 4^2 = 9 * 16 = 144

ex. (3

Division Rule: (x / y)^a =

x^a / y^a

ex. (2/3)^4 = 2^4 / 3^4 = 16 / 81

ex. (2/3)^4 = 2^4 / 3^4 = 16 / 81

Be able to manipulate each side of the equation using the rules stated to make both sides identical

ex. 4^3 = 2^x ; X=?

manipulate 4^3 such that matches 2^x format

(2^2)^3 = 2^6 = 2^x ; x=6

manipulate 4^3 such that matches 2^x format

(2^2)^3 = 2^6 = 2^x ; x=6

Multiple base manipulation

2^x ** 3^y ** 5^z = 2^5 ** 3^4 ** 5^5

x=5 ; y=4 ; z=5

x=5 ; y=4 ; z=5

When both sides of equation have identical Prime!! bases,

Then exponents must be equivalent to each other.

- No matter how many 2's are on a side, they can never combine with other numbers to result in a 3 or 5, or any other primes

- No matter how many 2's are on a side, they can never combine with other numbers to result in a 3 or 5, or any other primes

GOAL in EXPONENT equations

always try and make each side of equation contain the same prime numbers then equate exponents

75^y ** 27^2y+1 = 5^4 ** 3^x

(5**15)^y ** 27^2y+1 => 3^y ** 5^2y ** (3^3)^2y+1 =>

3^y** 5^2y ** 3^6y+3 => 5^2y ** 3^7y+3=5^4 ** 3^x

=2y=4 ; y=2 & 7y+3=x ; 14+3=x

3^y

=2y=4 ; y=2 & 7y+3=x ; 14+3=x

Adding & Subtracting Exponeents

*No specific rules

Key is you must rely on your understanding of factoring and combining like terms

Key is you must rely on your understanding of factoring and combining like terms

2^4x + 2^4x + 2^4x + 2^4x=4^24

4(2^4x) = 4^24 ; 2^2 * 2^4x => 2^4x+2=(2^2)^24

2^4x+2=2^48 ; 4x=46 or x=46/4 = 11.5

2^4x+2=2^48 ; 4x=46 or x=46/4 = 11.5

Common Exponents:

(-1)^2 = 1

0^2 = 0

1^2 = 1

0^2 = 0

1^2 = 1

2^2 =

4

2^3 =

8

2^4 =

16

2^5 =

32

2^6 =

64

3^2 =

9

3^3 =

27

3^4 =

81

4^2 =

16

4^3=

64

4^4 =

256

5^2 =

25

5^3 =

125

5^4 =

625

6^2 =

36

7^2 =

49

8^2 =

64

9^2 =

81

11^2 =

121

12^2 =

144

13^2 =

169

14^2 =

196

15^2 =

225

Roots are much like exponents

B/c all roots can be expressed as exponents;

W/ subtle differences on how to approach the problem

W/ subtle differences on how to approach the problem

Note: 4-root of 25 ^2 can be expressed as

(25^1/4)^2 = 25^2/4 = 25^1/2 =5

4th-root of 25^2 is same as

Taking 4th-root of 25, then ^2 it

1/2 root of 8 is same as

8^1/ 1/2 = 8^ 1/1*2/1 = 8^2

Factoring Roots

As long as all roots are the same, you can factor out

ex. sqrt(a/b = sqrt.a / sqrt.b

ex. sqrt(1/16 = sqrt.1 / sqrt.16 = 1/4

ex. sqrt(a/b = sqrt.a / sqrt.b

ex. sqrt(1/16 = sqrt.1 / sqrt.16 = 1/4

sqrt(a*b =

sqrt.a * sqrt.b

ex. sqrt(72 = sqrt(36**2 = 6**sqrt.2

ex. sqrt(72 = sqrt(36

When given unusual sqrt's, you might want to

Change to exponent form 1st; Then use exponent rules to make calculations necessary

Adding & Subtracting Roots

Works much like x+x+x+x=4x

sqrt2 + sqrt2 + sqrt2 + sqrt2=4*sqrt2

ex. sqrt2 + sqrt2 + sqrt8 = 2**sqrt2 + sqrt(4**2) = 2**sqrt2 + 2**sqrt2 = 4*sqrt2

sqrt2 + sqrt2 + sqrt2 + sqrt2=4*sqrt2

ex. sqrt2 + sqrt2 + sqrt8 = 2

CANNOT combine unlike terms =

sqrt12 + sqrt4 + sqrt16

It is not standard format to have the radical in the denominator

ex. 9 / sqrt3

so must multiply sqrt3 / sqrt3 by both sides = 9*sqrt3 / 3

3*sqrt3 / 1

so must multiply sqrt3 / sqrt3 by both sides = 9*sqrt3 / 3

3*sqrt3 / 1

3^ 3/2 =

3^1 + 3^1/2

*which is different from the way 3^ 2/3 breaksdown; = 3rd-Root of 3 ^2

*which is different from the way 3^ 2/3 breaksdown; = 3rd-Root of 3 ^2

Algebraic Number Properties (mostly dealing with exponents and roots)

...

Number property #1)

Any number raised to an even exponent you know the result is positive or zero; BUT you cannot know the sign of the number/variable

Number property #2)

When a number or variable is raised to an odd exponent, result can be positive, negative, or zero

ex. X^3 , y^5 -- (-4)^3 = -64

ex. X^3 , y^5 -- (-4)^3 = -64

Number property #3)

When a variable is raised to an odd exponent, the sign of the result determines the sign of the variable & will always be one solution

Number property #4)

When a number between 0&1 is squared; the result becomes smaller

-For all other real numbers greater than 1 or less than 0,(-), square of that number becomes greater

ex. (1/2)^2 = 1/4 ; (.4)^2 = .16 ; -(.4)^2 = -.16

-For all other real numbers greater than 1 or less than 0,(-), square of that number becomes greater

ex. (1/2)^2 = 1/4 ; (.4)^2 = .16 ; -(.4)^2 = -.16

Units Digits w/ exponents

There is a pattern for determining units digits for products & exponents.

Pattern Units Digits w/ exponents - "2" -

= 2, 4, 8, 6, 2,.....

Pattern Units Digits w/ exponents - "3" -

= 3, 9, 7, 1, 3,...

Pattern Units Digits w/ exponents - "4" -

= 4, 6, 4, 6,....

Pattern Units Digits w/ exponents - "5" -

= 5, 5, 5, 5,....

Pattern Units Digits w/ exponents - "6" -

= 6, 6, 6, 6,....

Pattern Units Digits w/ exponents - "7" -

= 7, 9, 3, 1, 7, 9....

Pattern Units Digits w/ exponents - "8" -

= 8, 4, 2, 6, 8, 4,....

Pattern Units Digits w/ exponents - "9" -

= 9, 1, 9, 1,.....

Root Number Property 1)

For even roots of all positive numbers, 2!!! solutions exist

One positive & One negative - However when sign for square root on gmat is used, only asking for positive/ or principal

- There is 1 solution for even root of 0

- An even root of a (-) number is not real & not on GMAT

One positive & One negative - However when sign for square root on gmat is used, only asking for positive/ or principal

- There is 1 solution for even root of 0

- An even root of a (-) number is not real & not on GMAT

Root Number Property 2)

Odd roots of all real numbers, there is exactly 1 solution

Solution can be positive, negative, or 0

Solution can be positive, negative, or 0

Root Number Property 3)

Taking the sqrt of a number between 0 & 1 results in a number greater than original.

- For all (+) numbers greater than 1, the sqrt of that number will be less than original

ex. sqrt of 1/4 = 1/2 & sqrt of 25 = 5

- For all (+) numbers greater than 1, the sqrt of that number will be less than original

ex. sqrt of 1/4 = 1/2 & sqrt of 25 = 5

Algebraic Calculations & Functions

Remember to factor, simplify, and combine like terms

Order of Operations -

P. E. M. D. A. S

Please Excuse My Dear Aunt Sally

Parentheses Exponents Multiply Divide Add Subtract

Please Excuse My Dear Aunt Sally

Parentheses Exponents Multiply Divide Add Subtract

Always "Why" and "How" to actively learn

Challenge yourself to get "it"; with everything

Key to Parentheses -

Avoid unecessary multiplying if a # will be used as/in a dividend later.

-Remember to:

Multiply every term within parentheses by number on outside

Multiply ever term within one set with every term in another(FOIL)

-Remember to:

Multiply every term within parentheses by number on outside

Multiply ever term within one set with every term in another(FOIL)

Key to Factoring -

Basically reversing parentheses rules; pull out common factors to form parentheses.

- With fractions, Factor both numerator and denominator individually 1st. Then look for like terms

Dont try and form & force like terms

- With fractions, Factor both numerator and denominator individually 1st. Then look for like terms

Dont try and form & force like terms

Linear Equation means -

All variables have an exponent of 1

Key to 1 variable equations -

Get the variable by itself by performing a series of operations (+, -, *, /). Always perform same operation on both sides

Key to multi-variable equations -

Generally will need (n-equations, for n # of variables)

However for exponent of small set problems, don't require full lot of equations to match

Typically will need to express 1 variable using the other variables in an equation. Then plug into other equations until able to solve.

However for exponent of small set problems, don't require full lot of equations to match

Typically will need to express 1 variable using the other variables in an equation. Then plug into other equations until able to solve.

Simultaneously solving by Adding/Subtracting both sides of 2 or more equations

ex.

x + y = 7

x - y = 1

_______

2X = 8 x =4, then plug 4 into either equation to solve for Y

x + y = 7

x - y = 1

_______

2X = 8 x =4, then plug 4 into either equation to solve for Y

Simultaneously solving more than 2 equations

Solve by putting 2 of the equations together to eliminate a variable. KEY is to eliminate a variable so, manipulate one or both equations all the way across to do so.

Then use that product to plug into other equations

Then use that product to plug into other equations

Quadratic Equations -

An equation that contains a squared variable & can not be solved by combining like terms &/or isolating unknown terms

Must be in: aX^2 + bX + c = 0

Must be in: aX^2 + bX + c = 0

Once quadratic equations are in proper form

You can factor by inspection or use quadratic formula

Quadratic Formula

x= -b +/- sqrt( b^2 - 4**a**c) / 2*a

has 2 solutions:

if b^2 - 4**a**c = 0, only one solution

if b^2 - 4**a**c < 0, then no solutions as can't have (-) sqrt

has 2 solutions:

if b^2 - 4

if b^2 - 4

Tougher equation to factor 3y^2 + 5y + 2 =0

(3y + _)(1y + _) =0

(3y + 2)(1y + 1) =0

y = -1 & -2/3

*Remember to set =0 and solve, so a x+1; is x = -1

(3y + 2)(1y + 1) =0

y = -1 & -2/3

*Remember to set =0 and solve, so a x+1; is x = -1

(x + y)^2 -breaks down into

x^2 + 2**x**y + y^2

(x - y)^2 -breaks down into

x^2 - 2**x**y + y^2

Difference of Squares = (x + y)(x - y)

x^2 - y^2

ex. (sqrt(2) + 1)(sqrt(2) - 1)(sqrt(3) + 1)(sqrt(3) - 1) =?

sqrt(2)^2 - 1^2 * sqrt(3)^2 - 1^2

2 -1) * (3-1 =

=2

2 -1) * (3-1 =

=2

Key Property of inequalities -

If you multiply or divide both sides by a negative number, the inequality flips

KEY!!! ex. ( X/Y > 3 = true? )

B/C you don't know the sign of y, you don't know if you need to flip inequality when multiplying both sides by Y. There is insufficient data to perform operation.

You must know whether y(or any variable) is (+) or (-)

You must know whether y(or any variable) is (+) or (-)

Inequality Fact - multiplying or dividing variables

You are not allowed to multiply or divide by a variable in an inequality, unless!!! you are sure of its sign

Inequality Fact - Subtracting?

DO NOT subtract inequalities to eliminate terms, b/c its the same as adding the 2nd equation after it has been multiplied by -1. Always try and add together

Inequality Fact - eliminating terms?

You are allowed to add 2 inequalities together to eliminate terms and solve for another term, AS LONG AS the sign's are pointing the same direction

Absolute Value and Inequalities create -

2 separate inequalities that consider 2 possible scenario's given by absolute value sign.

|x| < 5

1) x is positive/zero & x<5

2) x is negative & -x<5 or x>-5

which also combines to -5<x<5

|x| < 5

1) x is positive/zero & x<5

2) x is negative & -x<5 or x>-5

which also combines to -5<x<5

ex. |x| > 5

1) x >5

2) -x>5 = x<-5

can't combine so, either x >5 or x<-5

2) -x>5 = x<-5

can't combine so, either x >5 or x<-5

Memorize what |x| < y & |x| >y break down into

|x| < y = -y < x < y

|x| > y = x >y or x < -y

|x| > y = x >y or x < -y

When an absolute value includes an operation, |x-3|>5 breaks down into?

1) x-3>5 = x>8

2) -(x-3)>5 = -x+3>5; then *-1 across and = x-3<-5

= x< -2

2) -(x-3)>5 = -x+3>5; then *-1 across and = x-3<-5

= x< -2

Functions -

Another way to write an algebraic expression w/ 1 variable; x^2 + 5 is f=f(x) = x^2 + 5

-Think of the input of this function as "x" value & output of function as value defined by what x^2 + 5 yields when a value of x is filled in

-Think of the input of this function as "x" value & output of function as value defined by what x^2 + 5 yields when a value of x is filled in

Domain of a function -

is defined as the set of all allowable inputs for the function. Usually domain is set of all real numbers;

However when a sqrt(x+5) is in the denominator of a function, the domain is limited to value of x>5 b/c you can't have a "0" or negative square root

However when a sqrt(x+5) is in the denominator of a function, the domain is limited to value of x>5 b/c you can't have a "0" or negative square root

For problems asking about multiple functions -

You start with the outer function, and where ever there is an X, you replace with f(x) & solve for it.

Function Applications - Sequences -

Domain of a sequence(allowable input values) consists of positive integers. Then 1st term of a sequence is the output when input is =1; 2nd term of sequence is output when input is =2....

Fibonacci sequence

a of (n) = a of (n-2) + a of (n-1)