Probamos esto para el número entero $\orange n \geq 0$ y el número entero $\purple m \geq 0$:

$$\begin{array}{rcl}\blue a^\orange n \cdot \blue a^\purple m & =& \underbrace{\blue a \cdot \blue a \cdot \ldots \cdot \blue a}_{\orange n\textrm{ veces}}\cdot\underbrace{\blue a \cdot \blue a \cdot \ldots \cdot \blue a}_{\purple m\textrm{ veces}} \\ &=&\underbrace{\blue a \cdot \blue a \cdot \ldots \cdot \blue a}_{\orange n+\purple m\textrm{ veces}}\\&=&{\blue a}^{\orange n+\purple m}\end{array}$$

Ejemplo

$$\begin{array}{rcl}\blue{x}^\orange 2 \cdot \blue{x}^\purple 4 &=& \underbrace{\blue x \cdot \blue x}_{\orange 2\textrm{ veces}}\cdot\underbrace{\blue x \cdot \blue x \cdot \blue x \cdot \blue x}_{\purple 4\textrm{ veces}} \\ &=& \underbrace{\blue x \cdot \blue x \cdot \blue x \cdot \blue x \cdot \blue x \cdot \blue x}_{\orange 2+\purple 4\textrm{ veces}}\\&=& \blue{x}^{\orange 2+\purple 4}\\ \end{array}$$

Probamos esto para $\orange n\geq 0$:

$$\left(\frac{\blue a}{\green b}\right)^\orange n=\underbrace{\frac{\blue a}{\green b} \cdot \frac{\blue a}{\green b} \cdot \ldots \cdot \frac{\blue a}{\green b}}_{\orange n\textrm{ veces}}=\dfrac{\blue a^\orange n}{\green b^\orange n}\tiny.$$

Ejemplo

$$\begin{array}{c}\left(\dfrac{\blue{x}}{\green{y}}\right)^\orange 3 = \underbrace{\dfrac{\blue{x}}{\green{y}} \cdot \dfrac{\blue{x}}{\green{y}} \cdot \dfrac{\blue{x}}{\green{y}}}_{\orange 3\textrm{ veces}}= \dfrac{\blue{x}^\orange 3}{\green{y}^\orange3} \end{array}$$

Probamos esto para $\orange n\geq 0$:

$$\begin{array}{rcl}\left({\blue a}\cdot{\green b}\right)^{\orange n}&=&\underbrace{(\blue a \cdot \green b) \cdot (\blue a \cdot \green b) \cdot \ldots \cdot (\blue a \cdot \green b)}_{\orange n\textrm{ veces}} \\ &=& \underbrace{\blue a \cdot \blue a \cdot \ldots \cdot \blue a}_{\orange n\textrm{ veces}} \cdot \underbrace{\green b \cdot \green b \cdot \ldots \cdot \green b}_{\orange n\textrm{ veces}} \\&=& {\blue a^\orange n}\cdot{\green b^\orange n}\end{array}$$

Ejemplo

$$\begin{array}{rcl}\left(\blue{x}\cdot \green{y}\right)^{\orange 3} &=& \underbrace{\blue{x} \cdot \green{y} \cdot \blue{x} \cdot \green{y}\cdot \blue{x} \cdot \green{y}}_{\orange 3\textrm{ veces}} \\ &=& \underbrace{\blue x \cdot \blue x \cdot \blue x}_{\orange 3\textrm{ veces}} \cdot \underbrace{\green y \cdot \green y \cdot \green y}_{\orange 3\textrm{ veces}} \\&=& \blue{x}^\orange 3 \cdot \green{y}^\orange3 \end{array}$$

Probamos esto para $\orange n \geq 0$ y $\purple m \geq 0$:

$$\begin{array}{rcl}\left(\blue a^\orange n \right)^\purple m & =& \underbrace{\blue a^\orange n \cdot \blue a^\orange n \cdot \cdots \cdot \blue a^\orange n}_{\purple m\textrm{ veces}} \\ &=&\underbrace{\blue a \cdot \blue a \cdot \cdots \cdot \blue a}_{\orange n\cdot \purple m\textrm{ veces}}\\&=&\blue a^{\orange n\cdot \purple m}\end{array}$$

Ejemplo

$$\begin{array}{rcl}\left(\blue{x}^{\orange2}\right)^{\purple 3} &=& \underbrace{\blue{x}^\orange2 \cdot \blue{x}^\orange2 \cdot \blue{x}^\orange2}_{\purple 3 \text{ veces}} \\&=& \underbrace{\blue{x} \cdot \blue{x} \cdot \blue{x} \cdot \blue{x} \cdot \blue{x} \cdot \blue{x}}_{\orange 2 \cdot \purple 3 \text{ veces}} \\ &=& \blue{x}^{\orange2 \cdot \purple3} \end{array}$$