Algebraic Rules
a+b = b+a
ab = ba
( a + b) + c = a + (b + c )
( ab)c = a (bc )
Commutative Properties
Associative Properties
a+0 = 0+a = a
Additive Identity
Multiplication by Zero
a0 = 0a = 0
Multiplicative Identity
a1 = 1a = a
− a = (−1)a
Negative
Additive Inverse (Opposite)
a + (−a) = 0
Definition of Subtraction
a − b = a + ( −b )
Multiplicative Inverse (Reciprocal)
Distributive Property
a
1
= 1 for a ≠ 0
a
a (b + c ) = ab + ac
[So also, (b + c )a = ba + ca , a (b − c ) = ab − ac , and (b − c ) a = ba − ca ]
Rules for Fractions
P
Q PQ
=
R
R
PR P
=
QR Q
[Multiplying a number by a fraction.]
[Cancellation law, and used to change denominator.]
P R PR
=
Q S QS
P R P+R
+ =
Q Q
Q
a
is undefined.
0
P R PS
÷ =
Q S QR
P R P−R
− =
Q Q
Q
[Note: Denominators must be identical!]
0
= 0 for a ≠ 0 .
a
−a a
a
=
=−
−b
b
b
Exponents
an = a ⋅ a
a 0 = 1 for a ≠ 0
a for n = 1, 2,3,…
n times
a−n =
a n a m = a n+ m
an
= a n−m
am
1
an
(a n ) m = a nm
n
(ab) n = a nb n
an
⎛a⎞
=
⎜ ⎟
bn
⎝b⎠
[Note: Several uses of the
( a + b)(c + d ) = a (c + d ) + b(c + d ) = ac + ad + bc + bd
distributive property. This specific example is usually called “FOIL”.]
Some special products
(a + b) 2 = a 2 + 2ab + b 2
(a − b) 2 = a 2 − 2ab + b 2
a 3 + b3 = (a + b)(a 2 − ab + b 2 )
(a + b)(a − b) = a 2 − b 2
a 3 − b3 = (a − b)(a 2 + ab + b 2 )
The Addition Property of Equations
For any real number c, the equations a = b and a + c = b + c have the same solution
sets. [You can add or subtract the same value from both sides of an equation without
altering the solutions.]
The Multiplication Property of Equations
For any nonzero real number c, the equations a = b and ac = bc have the same
solution sets. [You can multiply or divide by any nonzero number on each side of an
equation without altering the solutions. Very useful in clearing fractions!]
Principle of Zero Products
If ab = 0 , then a = 0 or b = 0 (or both.)