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Rules of powers

A power $a^n$ is a shorthand for multiplying $a$ by itself $n$ times. When multiplying the same base the exponents are added, when dividing they are subtracted, and for a power of a power they are multiplied.

a^m\cdot a^n = a^{m+n}

multiplication: add the exponents

\frac{a^m}{a^n} = a^{m-n}

division: subtract the exponents

(a^m)^n = a^{mn}

power of a power: multiply the exponents

Symbols

$a$	base
$m,\ n$	exponents

Example

$2^3\cdot 2^4 = 2^7 = 128$ .

a^m\cdot a^n \neq a^{mn}

- the exponents are added, not multiplied.

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Part of Foundations of Mathematics: Powers, roots and scientific notation.