To simplify the expression $(3^2)^4$, you apply the power of a power rule, which states that when you raise a power to another power, you multiply the exponents. Thus, we can rewrite the expression as follows:

In this case, you start with the base $3$ raised to the exponent $2$, and then this entire expression is raised to the exponent $4$. According to the power of a power rule, you multiply the exponent $2$ by the exponent $4$:

Therefore, the expression $(3^2)^4$ simplifies to $3^8$.

To gain a deeper understanding of why this simplification is valid, let’s examine what $(3^2)^4$ represents. The expression $3^2$ means $3 \times 3$. When you raise this to the power of $4$, you are effectively multiplying $3^2$ by itself four times:

Since each $3^2$ equals $3 \times 3$, the complete multiplication can be expanded as follows:

Counting the total number of factors of $3$, you find that there are $8$ occurrences of $3$, which confirms that:

This exponentiation rule significantly aids in simplifying expressions and streamlining calculations, particularly when working with large numbers or more intricate algebraic expressions. Familiarizing yourself with the power of a power rule can not only save you time but also help minimize the possibility of errors in your mathematical computations.