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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^m-n (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 + (y2-y1)^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 Slope-Intercept Equation m is the slope, y ntercept is (0,b)
y - y1 = m(x - x1)
Point-Slope Equation  
Same slope with different y-intercepts.
Parallel Lines
m1m2 = -1
Perpendicular Lines Two lines with slopes m1 and m2 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 Simple-Interest 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 x-axis.
x-Axis Test of Symmetry
If replacing x with -x produces an equivalent equation, then the graph is symetric with respect to the y-axis.
y-Axis 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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