# Exponent Rules

### 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 = ( b) n 32 ⋅ 42 = (3⋅4)2 = 144
Quotient rules a n / a m = a nm 25 / 23 = 25-3 = 4
a n / b n = (/ 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 = nb 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

### 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 = ( b)n

Example:

32 ⋅ 42 = (3⋅4)2 = 122 = 12⋅12 = 144

### Exponents quotient rules

#### Quotient rule with same base

an / am = anm

Example:

25 / 23 = 25-3 = 22 = 2⋅2 = 4

#### Quotient rule with same exponent

an / bn = (/ 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

a nm a (nm)

Example:

232 = 2(32= 2(3⋅3) = 29 = 2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2 = 512

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