Summary Ultimate Algebra Cheat Sheet: Formulas, Rules, and Quick Problem-Solving Guide

Algebra Cheat Sheet


Basic Properties and Facts
Arithmetic Operations Properties of Inequalities
 
b ab If a < b then a + c < b + c and a − c < b − c
ab + ac = a (b + c) a =
c c
a a b
If a < b and c > 0 then ac < bc and <
b a a ac c c
=  =
c bc b b a b
c If a < b and c < 0 then ac > bc and >
c c
a c ad + bc a c ad − bc
+ = − = Properties
b d bd b d bd ( of Absolute Value
a if a ≥ 0
a−b b−a a+b a b |a| =
= = + −a if a < 0
c−d d−c c c c
a |a| ≥ 0 |−a| = |a|
ab + ac ad
= b + c, a 6= 0  cb  = a |a|
a bc |ab| = |a| |b| =
d b |b|

|a + b| 6 |a| + |b| Triangle Inequality
Exponent Properties
an am = an+m (ab)n = an bn
Distance Formula
(an )m = anm a0 = 1 , a 6= 0 If P1 = (x1 , y1 ) and P2 = (x2 , y2 ) are two
an 1  a n an points the distance between them is
m
= an−m = m−n =
bn
q
a a b
d (P1 , P2 ) = (x2 − x1 )2 + (y2 − y1 )2
n
 1
 n 1 1
am = am = (an ) m = an
a−n
Complex Numbers
 n √ √ √
 a −n b bn 1 i = −1 i2 = −1 −a = i a , a ≥ 0
= = n a−n = n
b a a a
(a + bi) + (c + di) = a + c + (b + d) i

Properties of Radicals (a + bi) − (c + di) = a − c + (b − d) i
√ 1 √n √ √
n
a = an ab = n a n b (a + bi) (c + di) = ac − bd + (ad + bc) i
√
(a + bi) (a − bi) = a2 + b2
r
p√
m n
√ a n
a
a= nm
a n
= √
b n
b √
√ |a + bi| = a2 + b2 Complex Modulus
n n
a = a if n is odd
(a + bi) = a − bi Complex Conjugate
√
n n
a = |a| if n is even
(a + bi) (a + bi) = |a + bi|2




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