108 terms

# Algebra Sec. 2

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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
x^5 =
= x*x*xxx*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 = (xy)^a+b
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
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
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
ex. 27^(2x+4) = 3^(3x+9); X=?
(3^3)^(2x+4) = 3^6x+12=3^3x+9
3x=-3 ; x=-1
Rule: (x*y)^a =
x^a * y^a
ex. (34)^2 = 3^2 4^2 = 9 * 16 = 144
Division Rule: (x / y)^a =
x^a / y^a
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
Multiple base manipulation
2^x 3^y 5^z = 2^5 3^4 5^5
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
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
(515)^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
*No specific rules
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
Common Exponents:
(-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
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
sqrt(a*b =
sqrt.a * sqrt.b
ex. sqrt(72 = sqrt(362 = 6sqrt.2
When given unusual sqrt's, you might want to
Change to exponent form 1st; Then use exponent rules to make calculations necessary
Works much like x+x+x+x=4x
sqrt2 + sqrt2 + sqrt2 + sqrt2=4*sqrt2
ex. sqrt2 + sqrt2 + sqrt8 = 2sqrt2 + sqrt(42) = 2sqrt2 + 2sqrt2 = 4*sqrt2
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
3^ 3/2 =
3^1 + 3^1/2
*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
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
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
Root Number Property 2)
Odd roots of all real numbers, there is exactly 1 solution
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
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
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)
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
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.
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
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
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
Once quadratic equations are in proper form
You can factor by inspection or use quadratic formula
x= -b +/- sqrt( b^2 - 4ac) / 2*a

has 2 solutions:
if b^2 - 4ac = 0, only one solution
if b^2 - 4ac < 0, then no solutions as can't have (-) sqrt
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
(x + y)^2 -breaks down into
x^2 + 2xy + y^2
(x - y)^2 -breaks down into
x^2 - 2xy + 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
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 (-)
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
ex. |x| > 5
1) 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
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
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
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