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:
a^{ n} = a × a × ... × a
n times
a is the base and n is the exponent.
Examples
3^{1} = 3
3^{2} = 3 × 3 = 9
3^{3} = 3 × 3 × 3 = 27
3^{4} = 3 × 3 × 3 × 3 = 81
3^{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} = 128 |
a^{ n} ⋅ b^{ n} = (a ⋅ b)^{ n} | 3^{2} ⋅ 4^{2} = (3⋅4)^{2} = 144 | |
Quotient rules | a^{ n} / a^{ m} = a^{ n}^{-m} | 2^{5} / 2^{3} = 2^{5-3} = 4 |
a^{ n} / b^{ n} = (a / b)^{ n} | 4^{3} / 2^{3} = (4/2)^{3} = 8 | |
Power 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} = 8 | |
b^{1/n} = ^{n}√b | 8^{1/3} = ^{3}√8 = 2 | |
Negative exponents | b^{-n} = 1 / b^{n} | 2^{-3} = 1/2^{3} = 0.125 |
Zero rules | b^{0} = 1 | 5^{0} = 1 |
0^{n} = 0 , for n>0 | 0^{5} = 0 | |
One rules | b^{1} = b | 5^{1} = 5 |
1^{n} = 1 | 1^{5} = 1 | |
Minus one rule | (-1)^{5} = -1 | |
Derivative rule | (x^{n})' = n⋅x^{ n}^{-1} | (x^{3})' = 3⋅x^{3-1} |
Integral rule | ∫ 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
See: 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^{-n} = 1 / b^{n}
Example:
2^{-3} = 1/2^{3} = 1/(2⋅2⋅2) = 1/8 = 0.125
See: Negative exponents
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