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Gimme all the rules of exponents with examples .
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Answers
The base a 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 ⋅ am = an+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 / am = an-m
Example:
25 / 23 = 25-3 = 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 = 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
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