The base a is raised to the power of n is equal to the multiplication of a, n times:
an = a × a × … × a
n times
a is the base and n is the exponent.
Exponents rules and properties
Rule name
Rule
Example
Product rules
a n · a m = a n+m
23 · 24 = 23+4 = 128
a n · b n = (a · b) n
32 · 42 = (3·4)2 = 144
Quotient rules
a n / a m = a n–m
25 / 23 = 25-3 = 4
a n / b n = (a / b) n
43 / 23 = (4/2)3 = 8
Power rules
(bn)m = bn·m
(23)2 = 23·2 = 64
bnm = b(nm)
232 = 2(32)= 512
m√(bn) = b n/m
2√(26) = 26/2 = 8
b1/n = n√b
81/3 = 3√8 = 2
Negative exponents
b-n = 1 / bn
2-3 = 1/23 = 0.125
Zero rules
b0 = 1
50 = 1
0n = 0 , for n>0
05 = 0
One rules
b1 = b
51 = 5
1n = 1
15 = 1
Minus one rule
(-1)5 = -1
Derivative rule
(xn)‘ = n·x n-1
(x3)‘ = 3·x3-1
Integral rule
∫ xndx = xn+1/(n+1)+C
∫ x2dx = x2+1/(2+1)+C
Exponents product rules
Product rule with same base
an · am = an+m
Example:
23 · 24 = 23+4 = 27 = 2·2·2·2·2·2·2 = 128
Product rule with same exponent
an · bn = (a · b)n
Example:
32 · 42 = (3·4)2 = 122 = 12·12 = 144
Exponents quotient rules
Quotient rule with same base
an / am = an–m
Example:
25 / 23 = 25-3 = 22 = 2·2 = 4
Quotient rule with same exponent
an / bn = (a / b)n
Example:
43 / 23 = (4/2)3 = 23 = 2·2·2 = 8
Exponents power rules
Power rule I
(an) m = a n·m
Example:
(23)2 = 23·2 = 26 = 2·2·2·2·2·2 = 64
Power rule II
anm = a(nm)
Example:
232 = 2(32) = 2(3·3) = 29 = 2·2·2·2·2·2·2·2·2 = 512
Power rule with radicals
m√(a n) = a n/m
Example:
2√(26) = 26/2 = 23 = 2·2·2 = 8
Negative exponents rule
b-n = 1 / bn
Example:
2-3 = 1/23 = 1/(2·2·2) = 1/8 = 0.125