Index laws
Laws that define how to deal with exponents in algebra.
Multiplication
am×an=am+n
Division
am÷an=am−n
Powers
(a4)3=a4×a4×a4
=a4+4+4
=a4×3=a12
(am)n=amn
(ab)n=anbn
xm+1×xm−1=x2m
Fractions
Since a21×a21=a1=a it follows that a21=2a.
anm=nam
a0.25=a41=4a
These may sometimes require to be simplified:
x1231=12x31
Since 12 goes into 31 two times, and the remainder is 7:
⇒12x24×12x7
And since the former factor simplifies further into x2, it follows that
12x31=x212x7
Zero Indices
an÷an=an−n=a0
So a0=an÷an, also, since every number divided by itself results in 1, it follows that a0=1.
00= undefined
Negative Indices
a−1 is equal to the reciprocal of a
a−n=an1
2a−3=a32
a−21=2a1
a−11=a
(x1)−1=1x=x
(ba)−c=acbc
Notes
Careful with negative numbers:
−42=(−4)2 because
−42=−(42)