Exponent rules

Exponent rules, laws of exponent and examples.

What is an exponent

The base a 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.

Examples

31 = 3

32 = 3 × 3 = 9

33 = 3 × 3 × 3 = 27

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34 = 3 × 3 × 3 × 3 = 81

35 = 3 × 3 × 3 × 3 × 3 = 243

310 = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 = 59 049

Exponents rules and properties

Exponents product rules

Product rule with same base

an ⋅ am = an+m

Example:

22 ⋅ 23 = 22+3 = 25 = 2⋅2⋅2⋅2⋅2 = 32

Product rule with same exponent

an ⋅ bn = (a ⋅ b)n

Example:

22 ⋅ 32 = (2⋅3)2 = 62 = 6⋅6 = 36

Exponents quotient rules

Quotient rule with same base

an / am = an-m

Example:

24 / 22 = 24-2 = 22 = 2⋅2 = 4

Quotient rule with same exponent

an / bn = (a / b)n

Example:

63 / 33 = (6/3)3 = 23 = 2⋅2⋅2 = 8

Exponents power rules

Power rule I

(an) m = a n⋅m

Example:

(21)3 = 21⋅3 = 23 = 2⋅2⋅2 = 8

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

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

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