Basics
$$ \begin{aligned}
a^n &= a \cdot a \cdot a \dots a\\[7pt]
\text{For } a \neq 0 \text{, we have:}\quad a^0 &= 1,\quad a^{-n}=\dfrac{1}{a^n}\\[7pt]
\text{For } a > 0 \text{, we have:}\quad a^b &= e^{b \cdot \ln(a)}
\end{aligned} $$
Powers with the Same Base
$$ \begin{aligned}
a^m \cdot a^n &= a^{(m+n)}\\[7pt]
\dfrac{a^m}{a^n} &= a^{(m-n)}
\end{aligned} $$
Examples:
$$ \begin{alignat}{7}
3^2 \cdot 3^4 &= a^{(2+4)} = 3^6 = 729\\[7pt]
\dfrac{5^4}{5^2} &= 5^{(4-2)} = 5^2 = 25
\end{alignat} $$
Powers with the Same Exponent
$$ \begin{aligned}
a^n \cdot b^n &= (a \cdot b)^n\\[7pt]
\dfrac{a^n}{b^n} &= \Bigl(\dfrac{a}{b}\Bigr)^n
\end{aligned} $$
Examples:
$$ \begin{aligned}
2^3 \cdot 4^3 &= (2 \cdot 4)^3 = 8^3 = 512\\[7pt]
\dfrac{8^5}{4^5} &= \Bigl(\dfrac{8}{4}\Bigr)^5=2^5=32
\end{aligned} $$
Raising Powers to Powers
$$ \begin{aligned}
\Bigl(a^m\Bigr)^n &= \Bigl(a^n\Bigr)^m =a^{(m \cdot n)}
\end{aligned} $$
Example:
$$ \begin{aligned}
\Bigl(2^4\Bigr)^3 &= 2^{(4 \cdot 3)} = 2^{12} = 4096
\end{aligned} $$
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