## Exponent rules

Exponent rules, laws of exponent and examples.

## What is an exponent

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

a=^{ n}a×a×...×an times

a is the base and n is the exponent.

## Examples

3

^{1}= 33

^{2}= 3 × 3 = 93

^{3}= 3 × 3 × 3 = 273

^{4}= 3 × 3 × 3 × 3 = 813

^{5}= 3 × 3 × 3 × 3 × 3 = 243## Exponents rules and properties

Rule name Rule Example Product rules a·^{ n}a=^{ m}a^{ n+m}2 ^{3}· 2^{4}= 2^{3+4}= 128a·^{ n}b= (^{ n}a·b)^{ n}3 ^{2}· 4^{2}= (3·4)^{2}= 144Quotient rules a/^{ n}a=^{ m}a^{ n}^{-m}2 ^{5}/ 2^{3}= 2^{5-3}= 4a/^{ n}b= (^{ n}a/b)^{ n}4 ^{3}/ 2^{3}= (4/2)^{3}= 8Power rules ( b)^{n}^{m}=b^{n·m}(2 ^{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}= 8b^{1/n}=√^{n}b8 ^{1/3}=^{3}√8 = 2Negative exponents b= 1 /^{-n}b^{n}2 ^{-3}= 1/2^{3}= 0.125Zero rules b^{0}= 15 ^{0}= 10 = 0 , for^{n}n>00 ^{5}= 0One rules b^{1}=b5 ^{1}= 51 = 1^{n}1 ^{5}= 1Minus one rule (-1) ^{5}= -1Derivative rule ( x)^{n}'=n·x^{ n}^{-1}( x^{3})'= 3·x^{3-1}Integral rule ∫ x=^{n}dxx^{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 = 144See: Multplying exponents

## 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 = 8See: Dividing exponents

## 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= 1 /^{-n}b^{n}Example:

2

^{-3}= 1/2^{3}= 1/(2·2·2) = 1/8 = 0.125See: Negative exponents

## See also