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Algebra Cheat Sheet

 

Number Rules

a·0=0 1·a=a


Expand Rules

(a±b)=ab a(b+c)=ab+ac
a(b+c)(d+e)=abd+abe+acd+ace (a+b)(c+d)=ac+ad+bc+bd
(a)=a


Fractions Rules

0a =0 , a0 a1 =a
aa =1 (ab )1=1ab  =ba 
(ab )c=((ab )1)c=(ba )c a1=1a 
ab=1ab  ab =ab 
ab =ab  ab =ab 
abc  =a·cb  bc a =bc·a 
1bc  =cb 


Absolute Rules

|a|=|a| |a|=a , a0
|ax|=a|x| , a0


Exponent Rules

1a=1 a1=a
a0=1 , a0 0a=0 , a0
(ab)n=anbn aman =amn , m>n
aman =1anm  , n>m ab+c=abac
(ab)c=ab·c abx=(ab)x
(ab )c=acbc  ac·bc=(a·b)c


Radical Rules

1=1 0=0
na=a1n  nam=amn 
aa=a nan=a, a0
nan=|a|, n is even nan=a, n is odd
nab=nanb, a,b0 nab =nanb , a,b0


Factor Rules

x2y2=(xy)(x+y)
x3+y3=(x+y)(x2xy+y2)
xnyn=(xy)(xn1+xn2y++xyn2+yn1)
xn+yn=(x+y)(xn1xn2y+xyn2+yn1)        n is odd
ax(2n)b=(axn+b)(axnb)
ax(4)b=(ax2+b)(ax2b)
ax(2n)by(2m)=(axn+bym)(axnbym)
ax(4)by(4)=(ax2+by2)(ax2by2)


Factorial Rules

n!(n+m)! =1(n+1)·(n+2)(n+m)  n!(nm)! =n·(n1)(nm+1),n>m
0!=1 n!=1·2(n2)·(n1)·n


Log Rules

log(1)=0 loga(a)=1
loga(xb)=b·loga(x) logab(x)=1b loga(x)
loga(1x )=loga(x) log1a (x)=loga(x)
loga(b)=ln(b)ln(a)  logx(xn)=n
logx((1x )n)=n aloga(b)=b


Undefined

00=Undefined x0 =Undefined
loga(b)=Undefined , a0 loga(b)=Undefined , b0
log1(a)=Undefined


Complex Number Rules

i2=1 1=i
a=1a