# 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 &amp; Exponents

= xxxxx

x

1

1 / x^a

x^a

x^a+b

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

y^a x^b = (xy)^a+b
b/c y &amp; x are different you can&#039;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&#039;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^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&#039;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 =&gt; 3^y 5^2y (3^3)^2y+1 =&gt;
3^y 5^2y 3^6y+3 =&gt; 5^2y 3^7y+3=5^4 3^x
=2y=4 ; y=2 &amp; 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 =&gt; 2^4x+2=(2^2)^24
2^4x+2=2^48 ; 4x=46 or x=46/4 = 11.5

(-1)^2 = 1
0^2 = 0
1^2 = 1

4

8

16

32

64

9

27

81

16

64

256

25

125

625

36

49

64

81

121

144

169

196

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&#039;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

...

### 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 &amp; will always be one solution

### Number property #4)

When a number between 0&amp;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 &amp; exponents.

### Pattern Units Digits w/ exponents - &quot;2&quot; -

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

### Pattern Units Digits w/ exponents - &quot;3&quot; -

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

### Pattern Units Digits w/ exponents - &quot;4&quot; -

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

### Pattern Units Digits w/ exponents - &quot;5&quot; -

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

### Pattern Units Digits w/ exponents - &quot;6&quot; -

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

### Pattern Units Digits w/ exponents - &quot;7&quot; -

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

### Pattern Units Digits w/ exponents - &quot;8&quot; -

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

### Pattern Units Digits w/ exponents - &quot;9&quot; -

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

### Root Number Property 1)

For even roots of all positive numbers, 2!!! solutions exist
One positive &amp; 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 &amp; 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 &amp; 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 &amp; sqrt of 25 = 5

### Algebraic Calculations &amp; 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 &quot;Why&quot; and &quot;How&quot; to actively learn

Challenge yourself to get &quot;it&quot;; 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 &amp; 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&#039;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 &amp; can not be solved by combining like terms &amp;/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 &lt; 0, then no solutions as can&#039;t have (-) sqrt

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

(3y + _)(1y + _) =0
(3y + 2)(1y + 1) =0
y = -1 &amp; -2/3
*Remember to set =0 and solve, so a x+1; is x = -1

x^2 + 2xy + y^2

x^2 - 2xy + y^2

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 &gt; 3 = true? )

B/C you don&#039;t know the sign of y, you don&#039;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&#039;s are pointing the same direction

### Absolute Value and Inequalities create -

2 separate inequalities that consider 2 possible scenario&#039;s given by absolute value sign.
|x| &lt; 5
1) x is positive/zero &amp; x&lt;5
2) x is negative &amp; -x&lt;5 or x&gt;-5
which also combines to -5&lt;x&lt;5

### ex. |x| &gt; 5

1) x &gt;5
2) -x&gt;5 = x&lt;-5
can&#039;t combine so, either x &gt;5 or x&lt;-5

### Memorize what |x| &lt; y &amp; |x| &gt;y break down into

|x| &lt; y = -y &lt; x &lt; y
|x| &gt; y = x &gt;y or x &lt; -y

### When an absolute value includes an operation, |x-3|&gt;5 breaks down into?

1) x-3&gt;5 = x&gt;8
2) -(x-3)&gt;5 = -x+3&gt;5; then *-1 across and = x-3&lt;-5
= x&lt; -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 &quot;x&quot; value &amp; 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&gt;5 b/c you can&#039;t have a &quot;0&quot; or negative square root

You start with the outer function, and where ever there is an X, you replace with f(x) &amp; 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)

Example: