A negative exponent "flips it"...............here are some rules........

Exponent rules

Exponent rules, laws of exponent and examples.

• What is an exponent
• Exponents rules
• Exponents calculator

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 × ... × a

                    n times

a is the base and n is the exponent.

Examples

3¹ = 3

3² = 3 × 3 = 9

3³ = 3 × 3 × 3 = 27

3⁴ = 3 × 3 × 3 × 3 = 81

3⁵ = 3 × 3 × 3 × 3 × 3 = 243

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 · a^m = a^n+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

See: Multplying exponents

Exponents quotient rules

Quotient rule with same base

aⁿ / a^m = a^n-m

Example:

2⁵ / 2³ = 2^5-3 = 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

See: Dividing exponents

Exponents power rules

Power rule I

(aⁿ) ^m = a ^n·m

Example:

(2³)² = 2^3·2 = 2⁶ = 2·2·2·2·2·2 = 64

Power rule II

_a n^m _= a (n^m)

Example:

₂3² _{= 2}(3²) _{= 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√(2⁶) = 2^6/2 = 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

See: Negative exponents

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