### Explanation

The **power of a power rule** is a property of exponents that simplifies expressions where one exponent is raised to another. This rule is particularly useful in algebra and calculus for simplifying complex exponential expressions.

### The Rule

In mathematical terms, the power of a power rule states that:

$(a^{m})^{n} = a^{m \cdot n}$
This means that when you have a base $a$ raised to an exponent $m$, and this entire expression is then raised to another exponent $n$, you can multiply the exponents $m$ and $n$ to get the simplified form.

### Example

Consider the expression $(3^{2})^{4}$:

$\begin{align*}
(3^{2})^{4} & = 3^{2 \cdot 4} \\
& = 3^{8}
\end{align*}$
So, $(3^{2})^{4}$ simplifies to $3^{8}$.

### Why It Works

To understand why this rule works, let's expand the expression $(a^{m})^{n}$ manually:

$\begin{align*}
(a^{m})^{n} & = a^{m} \times a^{m} \times \cdots \times a^{m} \quad (\text{n times}) \\
& = a^{m + m + \cdots + m} \quad (\text{n terms of } m) \\
& = a^{m \cdot n}
\end{align*}$
### Application

This rule is particularly useful in:

- Simplifying algebraic expressions
- Solving exponential equations
- Calculating higher-order derivatives in calculus where exponents are involved

By applying the power of a power rule, you can significantly reduce the complexity of calculations involving exponents, making it easier to manipulate and solve equations.