Laws of Exponents
Exponential expressions with integer, rational, or real exponents obey the following fundamental laws
- Product of powers with the same base
First law of exponents
The product of two powers sharing the same base is a single power with that base and an exponent equal to the sum of the exponents. $$ a^x \cdot a^y = a^{x+y} $$Example $$ 4^2 \cdot 4^3 = 4^{2+3} = 4^5 $$ Explanation $$ 4^2 \cdot 4^3 = (4 \cdot 4) \cdot (4 \cdot 4 \cdot 4) = 4^5 $$
- Quotient of powers with the same base
Second law of exponents
The quotient of two powers with the same base is a power with that base and an exponent given by the difference of the exponents, obtained by subtracting the denominator exponent from the numerator exponent. $$ \frac{a^x}{a^y} = a^{x-y}$$Example $$ \frac{5^4}{5^3}= 5^{4-3} = 5^1 $$ Explanation $$ \require{cancel} \frac{5^4}{5^3}= \frac{5 \cdot \cancel{5} \cdot \cancel{5} \cdot \cancel{5}}{\cancel{5} \cdot \cancel{5} \cdot \cancel{5}} = 5^1 $$
- Power of a power
Third law of exponents
Raising a power to another exponent yields a power with the same base and an exponent equal to the product of the exponents. $$ (a^x)^y = a^{x \cdot y}$$Example $$ (5^3)^2 = 5^{3 \cdot 2} = 5^6 $$ Explanation $$ (5^3)^2 = (5 \cdot 5 \cdot 5)^2 = (5 \cdot 5 \cdot 5)(5 \cdot 5 \cdot 5) = 5^6 $$
- Product of powers with the same exponent
Fourth law of exponents
The product of powers that share the same exponent is a power with that exponent and a base equal to the product of the bases. $$ a^x \cdot b^x = (a \cdot b)^x $$Example $$ 5^3 \cdot 2^3 = (5 \cdot 2)^3 = 10^3 $$ Explanation $$ 5^3 \cdot 2^3 = 5 \cdot 5 \cdot 5 \cdot 2 \cdot 2 \cdot 2 = (5 \cdot 2)(5 \cdot 2)(5 \cdot 2) = (5 \cdot 2)^3 = 10^3 $$
- Quotient of powers with the same exponent
Fifth law of exponents
The quotient of powers with the same exponent is a power with that exponent and a base equal to the quotient of the bases. $$ \frac{a^x}{b^x} = (\frac{a}{b})^x $$Example $$ \frac{8^3}{2^3} = (\frac{8}{2})^3 = 4^3 $$ Explanation $$ \frac{8^3}{2^3} = \frac{8 \cdot 8 \cdot 8}{2 \cdot 2 \cdot 2} = \frac{8}{2} \cdot \frac{8}{2} \cdot \frac{8}{2} = 4 \cdot 4 \cdot 4 = 4^3 $$
These laws follow directly from the definition of exponentiation as repeated multiplication. They formalize how multiplication and division act on exponential expressions.
Corollary
Several important corollaries extend these laws. The most fundamental is the following.
- Zero exponent law
Any nonzero number raised to the zero power equals 1. $$ \forall \ n \ne 0 \ \Rightarrow \ n^0 = 1 \ \ \ $$ See the proof.
Example $$ 4^0 = 1 $$
And so on.
