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a + b = b + a and ab = ba

Commutative properties of addition and multiplication.
 
a + (b + c) = (a + b) + c and a(bc) = (ab)c

Associative properties of addition and multiplication.
 
a + 0 = 0 + a = a

Additive identify property
 
a + a = a + (a) = 0

Additive inverse property
 
a * 1 = 1 * a = a

Multiplicative identity property
 
a * 1/a = 1/a * a = 1 (a ≠ 0)

Multiplicative inverse property
 
a(b + c) = ab + ac

Distributive Property
 
a = [a, if a ≥ 0.
[a, if a < 0.

Absolute Value
 
a^n = a * a * a * * * a,
 n factors 

Where a is the base and the n is the exponent.
 
a^m / a^n = a^mn (a ≠ 0)

Quotient Rule
 
a^0 = 1
and
a^m = 1/a^m

For any nonzero number a and any integer m.
 
a^m / b^n = b^n / a^m

For any nonzero numbers a and b and any integers m and n. (A factor can be moved to the other side of the fraction bar if the sign of the exponent is changed.
 
a^m * a^n = a^m+n

Product Rule
 
(a^m)^n = a^mn

Power Rule
 
(ab)^m = a^m * b^m

Raising a product to a power
 
(a / b)^m = a^m / b^m (b ≠ 0)

Raising a quotient to a power
 
( A + B)^2 = A^2 + 2AB + B^2

Square of a sum
 
( A  B)^2 = A^2  2AB + B^2

Square of a difference
 
( A + B)(A  B) = A^2  B^2

Product of a sum and a difference.
 
A^2  B^2 = (A + B)(A  B)

Difference of Squares
 
A^2 + 2AB + B^2 = (A +B)^2
A^2  2AB + B^2 = (A  B)^2

The rules for squaring binomials can be reversed to factor trinomials that are squares of binomials.
 
A^3 + B^3 = (A + B)(A^2  AB + B^2)
A^3  B^3 = (A  B)(A^2 + AB + B^2)

The following rules to a factor a sum or difference of cubes.
 
If a = b is true, then a + c = b + c is true.

The Addition Principle.
 
If a = b is true, then ac = bc is true.

The Multiplication Prinicple.
 
If ab = 0 is true, then a = 0 or b = 0, and if a = 0 or b = 0, then ab = 0.

The Principle of Zero Products
 
If x^2 = k, then x = √k or x = √k.

The Principle of Square Roots
 
1. If n is even, ^n√a^n = a
2. If n is odd, ^n√a^n = a 3. ^n√a * ^n√b = ^n√ab. 4. ^n√a/b = ^n√a / ^n√b (b ≠ 0) 5. ^n√a^m = (^n√a)^m 
Properties of Radicals
Let a and b be any real numbers or expressions for which the given roots exist. For any natural numbers m and n (n ≠ 1)
 
a^2 + b^2 = c^2

The Pythagorean Theorum
The sum of the squares of the lengths aof the legs of a right triangle is equal to the square of the length of the hypotenuse.
 
1. a^1/n = ^n√a,
2. a^m/n = ^n√a^m = (^n√a)^m
3. a^m/n = 1/a^m/n

Rational Exponents
For any real number a and any natural numbers m and n, n ≠ 1, for which ^n√a exists.
 
d = √(x2 x1)^2 + (y2y1)^2

The Distance Forumla
the distance d between any two points (x1,y1) and (x2,y2) is given by...
 
{[(x1 + x2)/2] , [(y1 + y2)/2]}

The Midpoint Forumla If the endpoijnts of a segment are (x1,y1) and (x2,y2), then the coordinates of the midpoint of the segment are...
 
(x  h)^2 + (y  k)^2 = r^2

The equation of a circle.
 
f(x) = mx + b

Linear Functions
A function of f is a linear function if it can be written as such... where m and b are constants. If m = 0, the function is a constant function f(x) = b. If m = 1 and b = 0, the function is the identity function f(x) = x.
 
m = rise/run
= the change in y/the change in x
= ((y2  y1)/(x2  x1) = (y1  y2)/(x1  x2))

The slope m of a line containing points (x1,y1) and (x2,y2) is given by... Slope is also considered an average rate of change.
 
f(x) = y = mx + b
or
f(x) = mx

The SlopeIntercept Equation
m is the slope, y ntercept is (0,b)
 
y  y_{1} = m(x  x_{1})

PointSlope Equation
 
Same slope with different yintercepts.

Parallel Lines
 
m_{1}m_{2} = 1

Perpendicular Lines
Two lines with slopes m_{1} and m_{2} are perpendicular if and only if the product of their slopes is 1.
Lines are also perpendicular if one is vertical (x = a) and the other is horizontal (y = b)
 
d = r · t

The Motion Forumla
The distance d traveled by an object moving at rate r in time t is given by...
 
I = Prt

The SimpleInterest Forumla
The simple interest I on a principal of P dollars at interest rate r for t years is given by...
 
(f + g)(x) = f(x) + g(x)

The Sum of Two Functions
 
(f  g)(x) = f(x)  g(x)

The Difference of Two Functions
 
(fg)(x) = f(x) · g(x)

The Product of Two Functions
 
(f/g)(x) = f(x)/g(x), g(x) ≠ 0

The Quotient of Two Functions
 
(f º g)(x) = f(g(x))

The Composition of Two Functions
 
If replacing y with y produces an equivalent equation, then the graph is symetric with respect to the xaxis.

xAxis Test of Symmetry
 
If replacing x with x produces an equivalent equation, then the graph is symetric with respect to the yaxis.

yAxis Test of Symmetry
 
If replacing x with x and y with y produces an equivalent equation, then the graph is symetric with respect to the origin.

Origin Test of Symmetry
 
f(x) = f(x)

Even Function
 
f(x) = f(x)

Odd Function
 
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