# Exponent Rules

The base a is raised to the power of n is equal to the multiplication of a, n times:

*a ^{n}* =

*a*×

*a*×

*…*×

*a*

n times

a is the base and n is the exponent.

Exponents rules and properties

**Rule name**

**Rule**

**Example**

*a*·

^{ n}*a*=

^{ m}*a*

^{ n+m}^{3}· 2

^{4}= 2

^{3+4}= 128

*a*·

^{ n}*b*= (

^{ n}*a*·

*b*)

^{ n}^{2}· 4

^{2}= (3·4)

^{2}= 144

*a*/

^{ n}*a*=

^{ m}*a*

^{ n}^{–m}

^{5}/ 2

^{3}= 2

^{5-3}= 4

*a*/

^{ n}*b*= (

^{ n}*a*/

*b*)

^{ n}^{3}/ 2

^{3}= (4/2)

^{3}= 8

*b*)

^{n}^{m}=

*b*

^{n·m}^{3})

^{2}= 2

^{3·2}= 64

_{b}n^{m}

_{= b}(

*n*

^{m})

_{2}3

^{2}

_{= 2}(3

^{2})

_{= 512}

^{m}√(

*b*) =

^{n}*b*

^{n/m}

^{2}√(2

^{6}) = 2

^{6/2}= 8

*b*

^{1/n}=

*√*

^{n}*b*

^{1/3}=

^{3}√8 = 2

*b*= 1 /

^{-n}*b*

^{n}^{-3}= 1/2

^{3}= 0.125

*b*

^{0}= 1

^{0}= 1

*= 0 , for*

^{n}*n*>0

^{5}= 0

*b*

^{1}=

*b*

^{1}= 5

*= 1*

^{n}^{5}= 1

^{5}= -1

*x*)

^{n}*‘*=

*n*·

*x*

^{ n}^{-1}

*x*

^{3})

*‘*= 3·

*x*

^{3-1}

*x*=

^{n}dx*x*

^{n}^{+1}/(

*n*+1)+

*C*

*x*

^{2}

*dx*=

*x*

^{2+1}/(2+1)+

*C*

### Exponents product rules

#### Product rule with same base

*a ^{n}* ·

*a*=

^{m}*a*

^{n+m}Example:

2^{3} · 2^{4} = 2^{3+4} = 2^{7} = 2·2·2·2·2·2·2 = 128

#### Product rule with same exponent

*a ^{n}* ·

*b*= (

^{n}*a*·

*b*)

^{n}Example:

3^{2} · 4^{2} = (3·4)^{2} = 12^{2} = 12·12 = 144

### Exponents quotient rules

#### Quotient rule with same base

*a ^{n}* /

*a*=

^{m}*a*

^{n}^{–m}

Example:

2^{5} / 2^{3} = 2^{5-3} = 2^{2} = 2·2 = 4

#### Quotient rule with same exponent

*a ^{n}* /

*b*= (

^{n}*a*/

*b*)

^{n}Example:

4^{3} / 2^{3} = (4/2)^{3} = 2^{3} = 2·2·2 = 8

### Exponents power rules

#### Power rule I

(*a ^{n}*)

^{ m}=

*a*

^{ n·m}Example:

(2^{3})^{2} = 2^{3·2} = 2^{6} = 2·2·2·2·2·2 = 64

#### Power rule II

_{a}n^{m} _{= }* _{a}*(

*n*

^{m})

Example:

_{2}3^{2} _{= 2}(3^{2}) _{= 2}(3·3) _{= 2}9_{ = 2·2·2·2·2·2·2·2·2 = 512}

#### Power rule with radicals

^{m}√(*a ^{ n}*) =

*a*

^{ n}^{/m}

Example:

^{2}√(2^{6}) = 2^{6/2} = 2^{3} = 2·2·2 = 8

### Negative exponents rule

*b ^{-n}* = 1 /

*b*

^{n}Example:

2^{-3} = 1/2^{3} = 1/(2·2·2) = 1/8 = 0.125